A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

Abstract

Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets 𝐽≡(𝐽,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets 𝐽 such that the symmetric Gram matrix 𝐺_𝐽∶=(1/2)[𝐶_𝐽+𝐶^tr_𝐽]∈𝕄_𝐽(ℚ) is positive semidefinite, where 𝐶_𝐽∈𝕄_𝐽(ℤ) is the incidence matrix of 𝐽. Following the idea of Drozd mentioned earlier, we associate to 𝐽 its Coxeter matrix Cox_𝐽∶=−𝐶_𝐽⋅𝐶^−tr_𝐽, its Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽, a Coxeter polynomial cox_𝐽(𝑡)∈ℤ[𝑡], and a Coxeter number𝐜_𝐽. In case 𝐺_𝐽 is positive semi-definite, we also associate to 𝐽 a reduced Coxeter number 𝐜̌_𝐽, and the defect homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ. In this case, the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of the unit circle and consists of roots of unity. In case 𝐺_𝐽 is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets 𝐽 with the Coxeter spectral properties of a simply laced Euclidean diagram 𝐷𝐽∈{𝔻^∼_𝑛,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} associated with 𝐽. Our aim of the Coxeter spectral analysis of such posets 𝐽 is to answer the question when the Coxeter type 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜̌_𝐽) of 𝐽 determines its incidence matrix 𝐶_𝐽 (and, hence, the poset 𝐽) uniquely, up to a ℤ-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any ℤ-invertible matrix 𝐴∈𝕄_𝑛(ℤ), there is 𝐵∈𝕄_𝑛(ℤ) such that 𝐴^tr=𝐵^tr⋅𝐴⋅𝐵 and 𝐵²=𝐸 is the identity matrix.

1. Introduction

In the present paper, we continue our Coxeter spectral study of finite posets, started in [1], in a close connection with the Coxeter spectral technique introduced in [2–4] for acyclic edge-bipartite graphs or signed graphs in the sense of [5]. We also follow some of the techniques of representation theory, graph combinatorics, and the spectral graph theory; see [6–31].

Here, we use the terminology and notation introduced in [1, 4, 26–28]. We denote by ℕ⊆ℤ⊆ℚ the set of nonnegative integers, the ring of integers, and the rational number field. Given 𝑚≥1, we view ℤ^𝑚 as a free abelian group and denote by 𝑒₁,…,𝑒_𝑚 the standard ℤ-basis of ℤ^𝑚. Given an index set 𝐽, we denote by ℤ^𝐽 the abelian group of all vectors 𝑣=(𝑣_𝑗)_𝑗∈𝐽, with integer coordinates 𝑣_𝑗∈ℤ, by 𝕄_𝐽(ℤ) the ℤ-algebra of all square 𝐽 by 𝐽 integral matrices, and by 𝐸∈𝕄_𝐽(ℤ) the identity matrix. In particular, 𝕄_𝑚(ℤ), with 𝑚≥1, is the ℤ-algebra of all square 𝑚 by 𝑚 matrices. The group Gl(𝑚,ℤ)∶=𝐴∈𝕄_𝑚(ℤ),det𝐴∈{−1,1}⊆𝕄_𝑚(ℤ) (1) is called the (integral) general linear group. We say that two square rational matrices 𝐴,𝐴^′∈𝕄_𝑚(ℚ) are ℤ-equivalent, or ℤ-congruent, (and denote 𝐴∼_ℤ𝐴^′) if there is a matrix 𝐵∈Gl(𝑚,ℤ) such that 𝐴^′=𝐵^tr⋅𝐴⋅𝐵. By a poset 𝐽≡(𝐽,⪯) we mean a finite partially ordered set 𝐽 with respect to a partial order relation ⪯. Following [26], a poset 𝐽 is called a one-peak poset if 𝐽 has a unique maximal element ∗. A finite poset 𝐽 is uniquely determined by its incidence matrix  𝐶_𝐽∈𝕄_𝐽(ℤ), that is, the square 𝐽×𝐽 matrix, as follows:𝐶_𝐽=𝑐_{𝑖𝑗𝑖,𝑗∈𝐽}∈𝕄_𝐽(ℤ),with𝑐_𝑖𝑗=1,for𝑖⪯𝑗,0,for𝑖⪯^╱𝑗. (2) Following an idea of Drozd [32] (developed in [27]), we have introduced in [1, 28] the Tits matrix  𝐶_𝐽∈𝕄_𝐽(ℤ) of 𝐽 to be the integral matrix 𝐶_𝐽=𝑐̂_{𝑖𝑗𝑖,𝑗∈𝐽}∈𝕄_𝐽(ℤ),with𝑐̂_𝑖𝑗1,𝑖=𝑗or𝑗⪯𝑖,𝑖,𝑗∉max𝐽,0,𝑖,𝑗incomparable,or𝑖⪯𝑗and𝑖,𝑗∉max𝐽,−1,if𝑖≺𝑗and𝑗∈max𝐽, (3) where max𝐽 is the set of all maximal elements of 𝐽. Usually, we equip the elements of 𝐽 with a numbering; that is, 𝐽 is viewed as 𝐽={𝑎₁,…,𝑎_𝑚}, 𝑚=|𝐽|≥1. Throughout, we fix such a numbering and make the identifications 𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) and ℤ^𝑚≡ℤ^𝐽. The incidence matrix 𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) and the Tits matrix 𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) depend on the numbering of 𝑎₁,…,𝑎_𝑚. Namely, if 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚} is obtained from 𝐽={𝑎₁,…,𝑎_𝑚} by a permutation 𝜎∈𝐒_𝑚 and 𝜎∈Gl(𝑚,ℤ) is the permutation matrix of 𝜎, then 𝐶_𝐽′=𝜎⁻¹⋅𝐶_𝐽⋅𝜎,𝐶_𝐽′=𝜎⁻¹⋅𝐶_𝐽⋅𝜎. (4) Note that any poset 𝐽 admits an upper-triangular numbering  𝐽={𝑎₁,…,𝑎_𝑚}; that is, 𝑎_𝑖⪯𝑎_𝑗 implies 𝑖≤𝑗. In this case, 𝐶_𝐽∈𝕄_𝑚(ℤ) is an upper-triangular matrix with 1 on the main diagonal, and, hence, det𝐶_𝐽=1, and det𝐶_𝐽′=1, for any numbering 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚}.

Fix a numbering 𝑎₁,…,𝑎_𝑚 of elements of 𝐽. Following [1, 28], by the Euler matrix of the poset 𝐽 we mean the inverse 𝐶_𝐽∶=𝐶_𝐽⁻¹∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) (5) of 𝐶_𝐽. Following [3, 4], we call 𝐴𝑑_𝐽∶=𝐶_𝐽+𝐶_𝐽^tr𝑃−2⋅𝐸,_𝐽(𝑡)=det𝑡⋅𝐸−𝐴𝑑_𝐽∈ℤ[𝑡] (6) the symmetric adjacency matrix and the characteristic polynomial of the poset 𝐽. The set 𝐬𝐩𝐞𝐜_𝐽 of all 𝑚=|𝐽| real roots of 𝑃_𝐽(𝑡) is defined to be the (real) spectrum of the poset 𝐽.

We denote by 𝑞_𝐽,𝑞_𝐽,𝑞_𝐽∶ℤ^𝐽≡ℤ^𝑚→ℤ the incidence quadratic form, the Tits quadratic form, and the Euler quadratic form of 𝐽 defined by the formulae 𝑞_𝐽(𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr=_𝑗∈𝐽𝑥²_𝑗+_𝑖≺𝑗𝑥_𝑖𝑥_𝑗,𝑞_𝐽(𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr=_𝑗∈𝐽𝑥²_𝑗+_{𝑖≺𝑗∈̌‌𝐽}𝑥_𝑖𝑥_𝑗−_{𝑝∈max𝐽𝑖≺𝑝}𝑥_𝑖𝑥_𝑝,𝑞_𝐽(𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr=𝑥⋅𝐶_𝐽⁻¹⋅𝑥^tr, (7) respectively, where 𝐽^̌‌=𝐽⧵max𝐽, max𝐽 is the set of all maximal elements in 𝐽, and 𝐶_𝐽∈𝕄_𝐽(ℤ) is the Tits matrix of 𝐽; see (27) and [1, 28] for a definition. The matrices 𝐺_𝐽1∶=2𝐶_𝐽+𝐶_𝐽^tr,𝐺_𝐽1∶=2𝐶_𝐽+𝐶_𝐽^tr,𝐺_𝐽1∶=2𝐶_𝐽+𝐶_𝐽^tr∈𝕄_𝐽(ℚ), (8) with rational coefficients, are called the symmetric incidence Gram matrix, the symmetric Tits-Gram matrix, and the symmetric Euler-Gram matrix of 𝐽. The matrices 𝐴𝑑_𝐽∶=𝐶_𝐽+𝐶_𝐽^tr−2⋅𝐸,𝐴𝑑_𝐽∶=𝐶_𝐽+𝐶_𝐽^tr−2⋅𝐸=𝐶_𝐽⁻¹+𝐶_𝐽^−tr−2⋅𝐸, (9) with integer coefficients, are called the Tits adjacency matrix, and the Euler adjacency matrix of 𝐽. The polynomials 𝑃_𝐽(𝑡)∶=det𝑡⋅𝐸−𝐴𝑑_𝐽=det𝑡⋅𝐸−𝐴𝑑_𝐽,𝑃_𝐽(𝑡)∶=det𝑡⋅𝐸−𝐴𝑑_𝐽 (10) are called the characteristic polynomial of 𝐽 and the Euler-characteristic polynomial of 𝐽, respectively.

Example 1. (a) If 𝐼 is the poset (11) then 𝑃_𝐼(𝑡)=𝑃_𝐼(𝑡)=𝑡⁴−5𝑡²−4𝑡; that is, the characteristic polynomial 𝑃_𝐼(𝑡) of 𝐼 coincides with the Euler-characteristic polynomial 𝑃_𝐼(𝑡) of 𝐼.
(b) If 𝐽 is the poset (12) of the Dynkin type 𝔼₆, then the characteristic polynomial 𝑃_𝐽(𝑡) of 𝐽 does not coincide with the Euler-characteristic polynomial 𝑃_𝐽(𝑡) of 𝐽, because 𝑃_𝐽(𝑡)=𝑡⁶−13𝑡⁴−26𝑡³−15𝑡²+2𝑡+3,𝑃_𝐽(𝑡)=𝑡⁶−5𝑡⁴+5𝑡²−1. (13)
Following [17, 33], we introduce the following definition.

Definition 2. (a) We define a poset 𝐽 to be positive (resp., nonnegative) if the incidence form 𝑞_𝐽∶ℤ^𝐽→ℤ of 𝐽 is positive (resp., nonnegative); that is, 𝑞_𝐽(𝑣)>0, for any nonzero 𝑣∈ℤ^𝐽 (resp., 𝑞_𝐽(𝑣)≥0, for any 𝑣∈ℤ^𝐽).
(b) We define a poset 𝐽 to be principal if its incidence form 𝑞_𝐽∶ℤ^𝐽→ℤ is principal in the sense of [34, Definition 2.1]; that is, 𝑞_𝐽 is nonnegative, not positive, and the kernel Ker𝑞_𝐽∶=𝑣∈ℤ^𝐽;𝑞_𝐽(𝑣)=0 (14) is an infinite cyclic subgroup of ℤ^𝐽.
Following the main idea of the Coxeter spectral analysis of acyclic edge-bipartite graphs (signed graphs) presented in [3, 4], we study finite posets 𝐽 (with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}) by means of the Coxeter spectrum𝐬𝐩𝐞𝐜𝐜_𝐽⊆ℂ (15) of 𝐽, that is, the set 𝐬𝐩𝐞𝐜𝐜_𝐽 of all 𝑚=|𝐽| eigenvalues of the Coxeter matrixCox_𝐽∶=−𝐶_𝐽⋅𝐶_𝐽^−tr∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) (16) of 𝐽, or equivalently, the set 𝐬𝐩𝐞𝐜𝐜_𝐽 of all 𝑚=|𝐽| roots of the Coxeter polynomialcox_𝐽(𝑡)∶=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽∈ℤ[𝑡]; (17) see (31) and [1]. It follows from (4) that the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 of 𝐽 and the spectrum 𝐬𝐩𝐞𝐜_𝐽 of 𝐽 do not depend on the numbering of the elements of the poset 𝐽.

A motivation. We recall from [26, 27] that the problems we study in the paper have a bimodule matrix problem interpretation and have essential applications in reducing some classes of partitioned matrices with coefficients in a field 𝐾 to their canonical forms. For simplicity of its presentation, we illustrate it in case when 𝑞_𝐽(𝑥) is the Tits quadratic form (7) of the poset 𝐽={𝑎₁,…,𝑎_𝑛,∗,+}, with an upper-triangular partial order ⪯ such that 𝐽 has precisely two maximal elements ∗∶=∗_𝑛+1 and +∶=+_𝑛+2. In this case, we have 𝑞_𝐽(𝑥)=_{𝑎𝑖∈𝐽}𝑥²_𝑖+_{𝑎𝑖≺𝑎𝑗,𝑖,𝑗≤𝑛}𝑥_𝑖𝑥_𝑗−_{𝑎𝑖≺∗}𝑥_𝑖𝑥_∗−_𝑎𝑗≺+𝑥_𝑗𝑥₊. (18) Fix a vector 𝑣=(𝑣₁,…,𝑣_𝑛,𝑣_∗,𝑣₊)∈ℕ^𝑛+2≡ℕ^𝐽, and consider the 𝐾-vector space 𝐌𝐚𝐭^𝐽_𝑣 of all partitioned matrices of the form (compare with [27]) (19) with coefficients in 𝐾, where 𝐴_𝑖∗=0 if 𝑎_𝑖≺∗̸ and 𝐴_𝑗+=0 if 𝑎_𝑗≺+̸. Consider the group 𝐆^𝐽_𝑣 generated by the elementary transformations of the following three types:(a)all simultaneous transformations on rows inside each horizontal block;(b)all simultaneous transformations on columns inside each vertical block;(c)all simultaneous transformations on columns from the 𝑖th column block to 𝑗th column block, if the relation 𝑎_𝑖⪯𝑎_𝑗 holds in the poset 𝐽⧵{∗,+} (with natural zero-adjustments).

It follows from [27, Section 2] (see also [16, 26, 32]) that the problem of finding canonical forms of matrices in 𝐌𝐚𝐭^𝐽_𝑣, with respect to the elementary transformations from the set 𝐆^𝐽_𝑣, is controlled by the Tits quadratic form 𝑞_𝐽 in the following sense. For any 𝑣∈ℕ^𝐽, there is only a finite number 𝐆^𝐽_𝑣-canonical forms of matrices in 𝐌𝐚𝐭^𝐽_𝑣 if and only if the form 𝑞_𝐽 is weakly positive; that is, 𝑞_𝐽(𝑣) is positive, for all nonzero vectors 𝑣∈ℕ^𝐽. Moreover, there is one-to-one correspondence between the irreducible 𝐆^𝐽_𝑣-canonical forms in 𝐌𝐚𝐭^𝐽_𝑣 and the vectors 𝑣∈ℕ^𝐽 satisfying the equation 𝑞_𝐽(𝑣)=1. A solution of the problem is given in [27] and [1, Theorem 1.6]. A useful homological interpretation (in terms of the Euler characteristic) of the ℤ-bilinear Tits form 𝑏̂_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr (26) and ℤ-bilinear Euler form 𝑏_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr is given in [1, (1.3)]. The reader is referred to [6–8, 25] for a detailed study and a solution of other important matrix problems of high computational complexity that have many useful applications in representation theory; see [16, 26].

We show in Section 3 that the Coxeter spectral analysis of principal posets 𝐽 essentially uses the Coxeter spectra of the simply laced Euclidean diagrams presented in Figure 1.

Figure 1 

Simply laced Euclidean (extended Dynkin) diagrams.

The nonsymmetric Gram matrix  𝐺^̌‌_Δ of any graph Δ=(Δ₀,Δ₁)∈{𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} of Figure 1, with the set of vertices Δ₀={1,…,𝑛,𝑛+1} and the set of edges Δ₁, is defined to be the matrix 𝐺^̌‌_Δ=1𝑑^Δ₁₂…𝑑^Δ_1𝑛𝑑^Δ_1𝑛+101…𝑑^Δ_2𝑛𝑑^Δ_2𝑛+1⋮⋮⋱⋮⋮00…1𝑑^Δ_𝑛𝑛+100…01∈𝕄_𝑛+1(ℤ), (20) where 𝑑^Δ_𝑖𝑗=−|Δ₁(𝑖,𝑗)|, if there is an edge •_𝑖---•_𝑗 and 𝑖≤𝑗. We set 𝑑^Δ_𝑖𝑗=0, if Δ₁(𝑖,𝑗) is empty or 𝑗<𝑖.

The Coxeter polynomialcox_Δ(𝑡)∶=det(𝑡⋅𝐸+𝐺^̌‌_Δ⋅𝐺^̌‌_Δ^−tr) of any diagram Δ=(Δ₀,Δ₁)∈{𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} does not depend on the numbering of the vertices in Δ₀ and is presented in (48). If 𝑛≥1 and Δ=𝔸^∼_𝑛, the Coxeter polynomialcox_Δ(𝑡)∶=det(𝑡⋅𝐸+𝐺^̌‌_Δ⋅𝐺^̌‌_Δ^−tr) of Δ depends on the numbering of the vertices in Δ₀ and is one of the polynomials 𝐹_Δ⁽¹⁾(𝑡),𝐹_Δ⁽²⁾(𝑡),…,𝐹^(𝑚𝑛)_Δ(𝑡) presented in [4], where 𝐹_Δ^(𝑗)(𝑡)=𝑡^𝑛+1−𝑡^{𝑛−𝑗+1}−𝑡^𝑗=()+1𝑡−1²⋅𝔳_𝑗(𝑡)⋅𝔳_{𝑛−𝑗+1}(𝑡),𝑚_𝑛=𝑛2,if𝑛iseven,𝑛+12,if𝑛+1iseven, (21) for 𝑗=1,…,𝑚_𝑛, and 𝔳_𝑚(𝑡)=𝑡^𝑚−1+𝑡^𝑚−2+⋯+𝑡²+𝑡+1. In particular, if 𝑛+1 is even and 𝑗=𝑚_𝑛=(𝑛+1)/2, then 𝑡^{𝑛−𝑗+1}=𝑡^𝑗 and 𝐹^(𝑚𝑛)_Δ(𝑡)=𝐹_Δ^((𝑛+1)/2)(𝑡)=𝑡^𝑛+1−2𝑡^(𝑛+1)/2+1. (22)

Following [4, 21], we associate (in Section 2) to any principal poset 𝐽 a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑛,𝑛≥1,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} such that the incidence symmetric Gram matrix 𝐺_𝐽∶=(1/2)[𝐶_𝐽+𝐶_𝐽^tr]∈𝕄_𝐽(ℚ) is ℤ-congruent to the symmetric Gram matrix𝐺_𝐷𝐽1∶=2^̌‌𝐺_𝐷𝐽+𝐺^̌‌tr_𝐷𝐽∈𝕄_𝐷𝐽(ℚ)=𝕄_𝐽(ℚ) (23) of 𝐷𝐽; that is, there is a ℤ-invertible matrix 𝐵 such that 𝐺_𝐷𝐽=𝐵^tr⋅𝐺_𝐽⋅𝐵.

One of the aims of the Coxeter spectral analysis of nonnegative finite posets is to study the question when the Coxeter type 𝐂𝐭𝐲𝐩𝐞_𝐽∶=𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜^̌‌_𝐽 (24) of a poset 𝐽 determines the matrix 𝐶_𝐽 (and, hence, the poset 𝐽) uniquely, up to a ℤ-congruency. Here, we set 𝐜^̌‌_𝐽=𝐜_𝐽, if 𝐽 is positive. In other words, we claim that, for any pair 𝐽, 𝐼 of nonnegative one-peak posets, 𝐂𝐭𝐲𝐩𝐞_𝐽=𝐂𝐭𝐲𝐩𝐞_𝐼 if and only if the incidence matrices 𝐶_𝐽 and 𝐶_𝐼 are ℤ-congruent. We also study the problem related with the results proved by Horn and Sergeichuk [35], if for any ℤ-invertible matrix 𝐴∈𝕄_𝑛(ℤ), there exists 𝐵∈𝕄_𝑛(ℤ) such that 𝐴^tr=𝐵^tr⋅𝐴⋅𝐵 and 𝐵²=𝐸 is the identity matrix; see [17, 18].

The main results of the present paper on nonnegative posets 𝐽 can be summarised as follows:

(1) canonical equivalences between the incidences, Tits, and Euler quadratic form (and corresponding Coxeter transformations and Coxeter spectra) of any poset 𝐽, established in Proposition 5;

(2) a characterization of principal posets given in Section 3. We show that a connected poset 𝐽 is principal if and only if there exists a simply laced Euclidean diagram 𝐷𝐽∈𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈ (25) such that the symmetric Gram matrix 𝐺_𝐽∶=(1/2)[𝐶_𝐽+𝐶_𝐽^tr]∈𝕄_𝐽(ℚ) of 𝐽 is ℤ-congruent to the symmetric Gram matrix 𝐺_𝐷𝐽∶=(1/2)[𝐺^̌‌_𝐷𝐽+𝐺^̌‌tr_𝐷𝐽]∈𝕄_𝐷𝐽(ℚ) of 𝐷𝐽. Moreover, we show in Section 3 that, given a connected principal poset 𝐽, the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of a unit circle 𝒮¹={𝑧∈ℂ;|𝑧|=1}, 1∈𝐬𝐩𝐞𝐜𝐜_𝐽, and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐽 is a root of unity;

(3) a Coxeter spectral classification result (Corollary 11) asserting that, given a pair 𝐼, 𝐽 of one-peak principal posets with at most 13 elements, the following conditions are equivalent:(3a)𝐷𝐼=𝐷𝐽,(3b)𝐬𝐩𝐞𝐜𝐜_𝐼=𝐬𝐩𝐞𝐜𝐜_𝐽,(3c)𝐜^̌‌_𝐼=𝐜^̌‌_𝐽 and |𝐼|=|𝐽|,(3d) the incidence matrix 𝐶_𝐽∈𝕄_𝐽(ℤ) is ℤ-congruent to the incidence matrix 𝐶_𝐼∈𝕄_𝐼(ℤ); that is, there is a ℤ-invertible matrix 𝐵 such that 𝐶_𝐼=𝐵^tr⋅𝐶_𝐽⋅𝐵.

In Section 3, we study principal posets by means of the defect and the reduced Coxeter number, and in Section 4, we present a framework for the study of nonnegative posets of corank 𝑟≥2 by means of their defect and the reduced Coxeter number. Examples are given in Sections 3–5.

The reader is referred to [1, 16, 17, 26] for a background of poset representation theory and elementary introduction to the poset matrix problems.

2. A Framework for the Coxeter Spectral Analysis of Finite Posets

The quadratic wanderings on finite posets 𝐽 studied in [1] are playing a key role in the representation theory of posets, algebras, and coalgebras, as well as in the algebraic combinatorics of posets; see [6, 9–14, 16, 24–26, 28, 31, 32, 36–39]. Except for the incidence wandering and the Euler wanderings defined by the incidence matrix 𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) (2), with det𝐶_𝐽=1 and a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}, as well as the Euler matrix 𝐶_𝐽∶=𝐶_𝐽⁻¹, we study in [1, 26–28] the Tits wandering defined by the Tits matrix  𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) of 𝐽 (see [28, (3.6)]), that is, the Gram matrix of the Tits  ℤ-bilinear form  𝑏̂_𝐽∶ℤ^𝐽×ℤ^𝐽→ℤ given by 𝑏̂_𝐽𝑥,𝑦=_{𝑎𝑖∈𝐽}𝑥_𝑖𝑦_𝑖+_{𝑎𝑗≺𝑎𝑖∈̌‌𝐽}𝑥_𝑖𝑦_𝑗−_{𝑎𝑝∈max𝐽𝑎𝑖≺𝑎𝑝}𝑥_𝑖𝑦_𝑝=𝑥⋅𝐶_𝐽⋅𝑦^tr, (26) where max𝐽 is the set of all maximal elements in the poset 𝐽 and 𝐽^̌‌∶=𝐽⧵max𝐽. We call 𝑞_𝐽(𝑥)∶=𝑏̂_𝐽(𝑥,𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr the Tits quadratic form of 𝐽.

A homological interpretation of the ℤ-bilinear forms 𝑏̂_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr and 𝑏_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr is given in [1, (1.3)]. For a geometric interpretation of the Tits form 𝑞_𝐼 of a one-peak poset 𝐼, the reader is referred to Drozd [32] and Simson [26].

Note that, given a one-peak poset 𝐼 of the form 𝐼={1,2,…,𝑛,∗=𝑛+1}, with a unique maximal element ∗=𝑛+1, we have 𝐶_𝐼=𝐶_𝑇^tr−𝑢01∈𝕄_𝑛+1(ℤ)1⋮1,with𝑢=, (27) where 𝐶_𝑇∈𝕄_𝑇(ℤ)=𝕄_𝑛(ℤ) is the incidence matrix of the poset 𝑇=𝐼⧵{∗}={1,2,…,𝑛}; see [26]. Note that 𝑞_𝐼(𝑥)=𝑥⋅𝐶_𝐼⋅𝑥^tr.

Now, we show that, in the Coxeter spectral study of finite posets 𝐽, we can use the Coxeter spectral technique introduced in [2, 4], for the edge-bipartite graphs (signed graphs [5]), and developed in [2, 34, 40] for the matrix morsifications of unit quadratic forms.

Following [3, 4, 24], by an edge-bipartite graph (bigraph, in short), we mean a pair Δ=(Δ₀,Δ₁), where Δ₀ is a finite nonempty set of vertices and Δ₁ is a finite set of edges equipped with a bipartition Δ₁=Δ⁻₁∪Δ⁺₁ such that the set Δ₁(𝑖,𝑗)=Δ⁻₁(𝑖,𝑗)∪Δ⁺₁(𝑖,𝑗) of edges connecting the vertices 𝑖 and 𝑗 does not contain edges lying in Δ⁻₁(𝑖,𝑗)∩Δ⁺₁(𝑖,𝑗), for each pair of vertices 𝑖,𝑗∈Δ₀, and either Δ₁(𝑖,𝑗)=Δ⁻₁(𝑖,𝑗) or Δ₁(𝑖,𝑗)=Δ⁺₁(𝑖,𝑗). Note that the edge-bipartite graphs can be viewed as signed multigraphs satisfying a separation property; see [4, 5].

We visualize Δ as a graph in a Euclidean space ℝ^𝑚, 𝑚≥2, with the vertices numbered by the integers 1,…,𝑛; usually, we simply write Δ₀={1,…,𝑛}. An edge in Δ⁻₁(𝑖,𝑗) is visualised as a continuous one •_𝑖---•_𝑗, and an edge in Δ⁺₁(𝑖,𝑗) is visualised as a dotted one •_𝑖---•_𝑗. A bigraph Δ is said to be loop-free if it has no loops.

We view any finite graph Δ=(Δ₀,Δ₁) as an edge-bipartite one by setting Δ⁻₁(𝑖,𝑗)=Δ₁(𝑖,𝑗) and Δ⁺₁(𝑖,𝑗)=∅, for each pair of vertices 𝑖,𝑗∈Δ₀.

To any loop-free edge-bipartite graph Δ=(Δ₀,Δ₁), with a fixed numbering Δ₀={𝑎₁,…,𝑎_𝑚} of its vertices, we associate the upper-triangular nonsymmetric Gram matrix  𝐺^̌‌_Δ=𝐸+[𝑑^Δ_𝑖𝑗]∈𝕄_𝑚(ℤ) of the form (20), with 𝑚∶=𝑛+1, where 𝑑^Δ_𝑖𝑗=−|Δ⁻₁(𝑖,𝑗)|, if there is an edge •_𝑖---•_𝑗 and 𝑖≤𝑗, 𝑑^Δ_𝑖𝑗=|Δ⁺₁(𝑖,𝑗)|, if there is an edge •_𝑖---•_𝑗 and 𝑖≤𝑗. We set 𝑑^Δ_𝑖𝑗=0, if Δ₁(𝑖,𝑗) is empty or 𝑗<𝑖. Since Δ is loop-free, we have 𝑑^Δ₁₁=⋯=𝑑^Δ_𝑚𝑚=0 and the main diagonal of 𝐺^̌‌_Δ consists of unities.

Following [4], we call Δ=(Δ₀,Δ₁)  positive (resp., nonnegative), if the symmetric Gram matrix 𝐺_Δ∶=(1/2)𝐺^̌‌_Δ+𝐺^̌‌_Δ^tr (28) of Δ is positive definite (resp., positive semidefinite).

Following [4], we associate to any loop-free edge-bipartite graph Δ, with |Δ₀|=𝑛≥2, the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_Δ⊆ℂ defined to be the spectrum of the Coxeter (-Gram) matrix Cox_Δ∶=−𝐺^̌‌_Δ⋅𝐺^̌‌_Δ^−tr∈𝕄_𝑛(ℤ), (29) the Coxeter polynomial cox_Δ(𝑡)∶=det𝑡⋅𝐸−Cox_Δ∈ℤ[𝑡], (30) the Coxeter transformation Φ_Δ∶ℤ^𝑛→ℤ^𝑛, given by 𝑥↦Φ_Δ(𝑥)∶=𝑥⋅Cox_Δ, the Coxeter number 𝐜_Δ (the order of Φ_Δ in the automorphism group of ℤ^𝑛, i.e., the minimal integer 𝑟≥1 such that Φ^𝑟_Δ=𝐸), the ℤ-bilinear Gram form 𝑏_Δ∶ℤ^𝑛×ℤ^𝑛→ℤ of Δ given by 𝑏_Δ(𝑥,𝑦)∶=𝑥⋅𝐺^̌‌_Δ⋅𝑦^tr, and the integral unit quadratic form 𝑞_Δ(𝑥)∶=𝑏_Δ(𝑥,𝑥)=𝑥²₁+⋯+𝑥²_𝑛+_𝑖<𝑗𝑑^Δ_𝑖𝑗𝑥_𝑖𝑥_𝑗=𝑥⋅𝐺_Δ⋅𝑥^tr. (31) Conversely, following Ovsienko [24], to any integral unit form 𝑞(𝑥)=𝑥²₁+⋯+𝑥²_𝑛+_𝑖<𝑗𝑞_𝑖𝑗𝑥_𝑖𝑥_𝑗,with𝑞_𝑖𝑗∈ℤ, (32) we associate the loop-free bigraph 𝐛𝐢𝐠𝐫(𝑞) of 𝑞 as follows (see also [34, 41]):(a)the vertices of 𝐛𝐢𝐠𝐫(𝑞) are the integers 1,…,𝑛,(b)two vertices 𝑖≠𝑗 are joined by −𝑞_𝑖𝑗 continuous edges of the form •_𝑖---•_𝑗 if 𝑞_𝑖𝑗 is negative, and by 𝑞_𝑖𝑗 dotted edges of the form •_𝑖---•_𝑗, if 𝑞_𝑖𝑗 is positive,(c)there is no edge between 𝑖 and 𝑗, if 𝑞_𝑖𝑗=0, or 𝑖=𝑗.

To any poset 𝐽≡(𝐽,⪯), with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚} of its points, we associate the following three edge-bipartite graphs: Δ_𝐽∶=𝐛𝐢𝐠𝐫𝑞_𝐽,Δ_𝐽∶=𝐛𝐢𝐠𝐫𝑞_𝐽,Δ_𝐽∶=𝐛𝐢𝐠𝐫𝑞_𝐽, (33) where 𝐛𝐢𝐠𝐫(𝑞_𝐽), 𝐛𝐢𝐠𝐫(𝑞_𝐽), and 𝐛𝐢𝐠𝐫(𝑞_𝐽) are the bigraphs of the quadratic forms 𝑞_𝐽, 𝑞_𝐽, and 𝑞_𝐽, respectively; see (7). More precisely, the bigraphs (33) are defined as follows.(i)The set of vertices of each of the bigraphs Δ_𝐽, Δ_𝐽, and Δ_𝐽 is the enumerated set 𝐽={𝑎₁,…,𝑎_𝑚}.(ii)There is an edge 𝑎_𝑖---𝑎_𝑗 in Δ_𝐽, if 𝑎_𝑖≺𝑎_𝑗 or 𝑎_𝑗≺𝑎_𝑖 holds in 𝐽.(iii)There is an edge 𝑎_𝑖---𝑎_𝑗 in Δ_𝐽, if 𝑎_𝑖 and 𝑎_𝑗 are not maximal in 𝐽 and 𝑎_𝑖≺𝑎_𝑗 or 𝑎_𝑗≺𝑎_𝑖 holds in 𝐽. There is an edge 𝑎_𝑖---𝑎_𝑗 in Δ_𝐽, if 𝑎_𝑖≺𝑎_𝑗 holds and 𝑎_𝑗 is maximal in 𝐽.(iv)Let 𝐶_𝐽=𝐶_𝐽⁻¹=[𝑐_𝑖𝑗]∈𝕄_𝑚(ℤ) be the Euler matrix of 𝐽. There is an edge 𝑎_𝑖---𝑎_𝑗 (resp., 𝑎_𝑖---𝑎_𝑗) in Δ_𝐽, if 𝑐_𝑖𝑗>0 or 𝑐_𝑗𝑖>0 (resp., 𝑐_𝑖𝑗<0 or 𝑐_𝑗𝑖<0).

We call Δ_𝐽, Δ_𝐽, and Δ_𝐽 the incidence bigraph of  Δ, the Tits bigraph of   Δ, and the Euler bigraph of  Δ, respectively, (with respect to the numbering 𝐽={𝑎₁,…,𝑎_𝑚}).

The following simple lemma is of importance.

Lemma 3. Assume that 𝐽 is a finite poset with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}, and let Δ_𝐽, Δ_𝐽, Δ_𝐽 be the loop-free edge-bipartite graphs associated with 𝐽 in (33).(a)The symmetric Gram matrices 𝐺_𝐽, 𝐺_𝐽, 𝐺_𝐽 are ℤ-congruent to the symmetric Gram matrices 𝐺_Δ𝐽, 𝐺Δ_𝐽, 𝐺_Δ𝐽, respectively. The rank of each of the symmetric Gram matrices 𝐺_Δ𝐽, 𝐺Δ_𝐽, 𝐺_Δ𝐽 does not depend of the numbering 𝐽={𝑎₁,…,𝑎_𝑚} and coincides with the common rank rk𝐺_Δ𝐽Δ=rk𝐺_𝐽=rk𝐺_Δ𝐽.(b)𝑃_𝐽(𝑡)=𝑃_Δ𝐽Δ(𝑡)=𝑃_𝐽(𝑡).(c)The poset 𝐽 is positive (resp., nonnegative) if and only if the bigraph Δ_𝐽  (and Δ_𝐽, Δ_𝐽) is positive (resp., nonnegative).(d)The poset 𝐽 is principal if and only if the bigraph Δ_𝐽  (and Δ_𝐽, Δ_𝐽) is principal.

Proof. For the proof of (a), we recall that the Gram matrices 𝐺_𝐽, 𝐺_𝐽, 𝐺_𝐽, 𝐺_Δ𝐽, 𝐺Δ_𝐽, 𝐺_Δ𝐽 are invariant, up to ℤ-congruency, under permutations of the elements {𝑎₁,…,𝑎_𝑚}. Since 𝐽 admits an upper-triangular numbering 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚} and 𝐺_Δ𝐽′=𝐺_𝐽′, then (a) follows. The proof of remaining statements is left to the reader.

Following the terminology used in [2–4, 34], we introduce the following definition.

Definition 4. Let 𝐽 be a finite poset, with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}.(a) We associate with 𝐽 the following three Coxeter matrices:(a1) the (incidence) Coxeter matrix Cox_𝐽∶=−𝐶_𝐽⋅𝐶_𝐽^−tr∈𝕄_𝑚(ℤ);(a2) the Coxeter-Tits matrix Cox_𝐽∶=−𝐶_𝐽⋅𝐶_𝐽^−tr∈𝕄_𝑚(ℤ);(a3) the Coxeter-Euler matrix Cox_𝐽∶=−𝐶_𝐽⁻¹⋅𝐶_𝐽^tr∈𝕄_𝑚(ℤ). Moreover, we define the following three Coxeter transformations:(a4) the (incidence) Coxeter transformation Φ_𝑚∶ℤ^𝑚→ℤ^𝑚 of 𝐽;(a5) the Coxeter-Tits transformation Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 of 𝐽;(a6) the Coxeter-Euler transformation Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 of 𝐽, by the following formulae: Φ_𝐽(𝑥)=𝑥⋅Cox_𝐽,Φ_𝐽(𝑥)=𝑥⋅Cox_𝐽,andΦ_𝐽(𝑥)=𝑥⋅Cox_𝐽. (34)(b) The integral polynomial cox_𝐽(𝑡)=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽∈ℤ[𝑡] (35) is called the Coxeter polynomial of the poset 𝐽.(c) The Coxeter spectrum of 𝐽 is the set 𝐬𝐩𝐞𝐜𝐜_𝐽⊆ℂ of all 𝑚=|𝐽| eigenvalues of the matrix Cox_𝐽, or, equivalently, the set 𝐬𝐩𝐞𝐜𝐜_𝐽 of all 𝑚=|𝐽| roots of the Coxeter polynomial cox_𝐽(𝑡).(d) The order 𝐜_𝐽∶=ord(Φ_𝐽) of the Coxeter transformation Φ_𝐽∶ℤ^𝐽→ℤ^𝐽 is called the Coxeter number of the poset 𝐽. In other words, 𝐜_𝐽 is the minimal integer 𝑟≥1 such that Φ^𝑟_𝐽=𝑖𝑑. We set 𝐜_𝐽=∞, if Φ^𝑟_𝐽≠𝑖𝑑, for any 𝑟≥1.(e) Assume that 𝐽 is nonnegative. The Coxeter type of 𝐽 is defined to be the pair 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽) if 𝐽 is positive, and the triple 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜^̌‌_𝐽) if 𝐽 is not positive, where 𝐜^̌‌_𝐽 is the reduced Coxeter number of 𝐽 in the sense of Theorems 10 and 18.
The following proposition shows that equality (35) holds.

Proposition 5. Let 𝐽 be a finite poset, with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}, let 𝑞_𝐽,𝑞_𝐽,𝑞_𝐽∶ℤ^𝑚→ℤ be the incidence, Tits, and Euler quadratic form of 𝐽, and let Φ_𝐽,Φ_𝐽,Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 be the corresponding Coxeter transformations.(a) The following equalities hold 𝐶_𝐽=𝐵⋅𝐶_𝐽⋅𝐵^𝑡𝑟 and 𝐶_𝐽^𝑡𝑟=𝐵^′⋅𝐶_𝐽⋅𝐵^′𝑡𝑟, and the following diagrams are commutative (36) where 𝐵^′=𝐶_𝐽^𝑡𝑟, 𝐶𝐵=^̌‌𝐽00𝐸, 𝐽^̌‌=𝐽⧵max𝐽, and ℎ_𝐵,ℎ_𝐵′∶ℤ^𝑚→ℤ^𝑚 are the group isomorphisms defined by the formulae ℎ_𝐵(𝑥)=𝑥⋅𝐵 and ℎ_𝐵′(𝑥)=𝑥⋅𝐵^′, for 𝑥∈ℤ^𝑚.(b)Cox_𝐽=𝐵⋅Cox_𝐽⋅𝐵⁻¹, Cox_𝐽⁻¹=Cox_𝐽^op=𝐵^′⋅Cox_𝐽⋅𝐵^′−1, and Φ_𝐽^op=Φ_𝐽⁻¹.(c) The Coxeter number 𝐜_𝐽=ord(Φ_𝐽) of the poset 𝐽 coincides with the Coxeter number of 𝐽^op. Moreover, 𝐜_𝐽=ord(Φ_𝐽)=ord(Φ_𝐽) and cox_𝐽^op(𝑡)=cox_𝐽(𝑡).(d) Assume that 𝐽 is connected and nonnegative.(d1) If the numbering 𝐽={𝑎₁,…,𝑎_𝑚} is upper-triangular and Δ_𝐽 is the bigraph (33) associated to 𝐽, then Cox_Δ𝐽=Cox_𝐽 and cox_Δ𝐽(𝑡)=cox_𝐽(𝑡).(d2) The Coxeter type 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜^̌‌_𝐽) of 𝐽 does not depend on the numbering 𝐽={𝑎₁,…,𝑎_𝑚}.(d3) The Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of a unit circle 𝒮¹={𝑧∈ℂ;|𝑧|=1}, and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐽 is a root of unity.(d4) The poset 𝐽 is positive if and only if 1∉𝐬𝐩𝐞𝐜𝐜_𝐽.

Proof. The first equality 𝐶_𝐽=𝐵⋅𝐶_𝐽⋅𝐵^tr is obvious, and the second one 𝐶_𝐼^tr=𝐵^′⋅𝐶_𝐽⋅𝐵^'tr follows by a direct calculation. Hence, (b) follows and, consequently, the diagrams (36) are commutative; see [1, Proposition 3.13]. Hence, the statement (c) follows from the commutativity of the diagrams (36).(d1) We recall from Section 1 that, given two numberings 𝐽={𝑎₁,…,𝑎_𝑚} and 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚} of elements in 𝐽, we have 𝐶_𝐽′=𝜎⁻¹⋅𝐶_𝐽⋅𝜎, where 𝜎∈Gl(𝑚,ℤ) is the permutation matrix of a permutation 𝜎∈𝐒_𝑚. Hence, (d1) easily follows.(d2) It is sufficient to note that the incidence matrix 𝐶_𝐽 is upper triangular. Hence, 𝐶_𝐽=𝐺^̌‌_Δ𝐽 and Cox_Δ𝐽=Cox_𝐽.
To prove (d3) and (d4), we recall from [2] and [3, Proposition 2.6] that the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐴 of any matrix morsification 𝐴∈𝐌𝐨𝐫_Δ of a nonnegative bigraph Δ is a subset of the unit circle 𝒮¹ and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐴 is a root of unity (see also [41, 42]). Moreover, Δ is positive iff 1∉𝐬𝐩𝐞𝐜𝐜_𝐴. Assume that 𝐽 is connected and nonnegative. Then, the bigraph Δ_𝐽 (33) associated to 𝐽 is nonnegative, 𝐴∶=𝐺^̌‌_Δ𝐽=∇(𝐶_𝐽) is a morsification of Δ_𝐽, and 𝐬𝐩𝐞𝐜𝐜_𝐽=𝐬𝐩𝐞𝐜𝐜_𝐴, because the incidence matrix 𝐶_𝐽 is quasitriangular and [4, Proposition 2.2] applies. This completes the proof.

Corollary 6. For any poset 𝐽, equality (35) holds.

Proof. Apply Proposition 5(b).

The following example shows that the correspondence 𝐽↦Δ_𝐽 defined in (33) does not preserve the Coxeter types of 𝐽 and Δ_𝐽. In particular, it shows that the equality cox_𝐽(𝑡)=cox_Δ𝐽(𝑡) does not hold in general and the Coxeter polynomial cox_Δ𝐽(𝑡) depends on the numbering of 𝐽, whereas the Coxeter polynomial cox_𝐽(𝑡) does not depend on the numbering of 𝐽.

Example 7. Consider the poset 𝐽 such that its Hasse quiver has the form (37) By a permutation of the elements in 𝐽, we get (38) Note that the first numbering is upper-triangular, whereas the second one is not upper-triangular.

3. Principal Posets

We recall that a poset 𝐽 is principal if its incidence unit form 𝑞_𝐽 is principal in the sense of [34, Definition 2.1]; that is, 𝑞_𝐽∶ℤ^𝐽→ℤ is nonnegative and not positive, and the kernel Ker𝑞_𝐽∶={𝑣∈ℤ^𝐽;𝑞_𝐽(𝑣)=0} is an infinite cyclic subgroup of ℤ^𝐽.

We start with the following useful observation.

Lemma 8. Assume that 𝐽 is a connected principal poset.(a) The Coxeter number 𝐜_𝐽 of 𝐽 is infinite.(b) The Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of a unit circle 𝒮¹={𝑧∈ℂ;|𝑧|=1}, 1∈𝐬𝐩𝐞𝐜𝐜_𝐽, and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐽 is a root of unity.(c) If Ker𝑞_𝐽=ℤ⋅𝐡, then Ker𝑞_𝐽=ℤ⋅𝐡 and Ker𝑞_𝐽=ℤ⋅𝐡, where(i)𝐡=𝐡⋅𝐵^′, 𝐡=𝐡⋅𝐵, 𝐡=𝐡⋅𝐵^′⋅𝐵⁻¹,(ii)𝐵^′=𝐶_𝐽^tr, 𝐶𝐵=^̌‌𝐽00𝐸, and 𝐽^̌‌=𝐽⧵max𝐽 are as in Proposition 5.

Proof. (a) By Proposition 5(d2), 𝐜_𝐽 is independent of the numbering of 𝐽. Then, without loss of generality, we may suppose that the numbering of 𝐽 is upper-triangular. Then, by Lemma 3(d) and Proposition 5(d1), the Coxeter number 𝐜_𝐽 coincides with the Coxeter number of the principal edge-bipartite graph Δ_𝐽 associated with 𝐽 in (33). Then, (a) is a consequence of [3, Proposition 3.12].
The statements (b) and (c) follow by applying Proposition 5 and the commutative diagram (36).

Proposition 9. Let 𝐽 be a connected poset, 𝑚=|𝐽|≥2, and let 𝐺_𝐽,𝐺_𝐽,𝐺_𝐽,∈𝕄_𝐽(ℚ) be the symmetric incidence Gram matrix of 𝐽, the symmetric Tits-Gram matrix of 𝐽, and the symmetric Euler-Gram matrix of 𝐽, respectively. The following five conditions are equivalent.(a)The poset 𝐽  is principal.(b)The Gram matrix 𝐺_𝐽 is positive indefinite of rank 𝑚−1.(c)The Tits quadratic form 𝑞_𝐽 of 𝐽 is nonnegative and Ker𝑞_𝐽=ℤ⋅𝐡, for some nonzero vector 𝐡∈ℤ^𝐽. (d)The Euler quadratic form 𝑞_𝐽 of𝐽 is nonnegative and  Ker𝑞_𝐽=ℤ⋅𝐡, for some nonzero vector 𝐡∈ℤ^𝐽.(e)If 𝐺 is any of the symmetric Gram matrices 𝐺_𝐽,𝐺_𝐽,𝐺_𝐽,∈𝕄_𝐽(ℚ) of 𝐽, then there exists a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑠,𝑠≥3,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈}  (uniquely determined by 𝐽) such that the matrix 𝐺 is ℤ-congruent to the symmetric Gram matrix 𝐺_𝐷𝐽∈𝕄_𝐷𝐽(ℚ) of the Euclidean diagram 𝐷𝐽; that is, there is a ℤ-invertible matrix 𝐵∈Gl(𝑚,ℤ) such that𝐺_𝐷𝐽=𝐵^tr⋅𝐺⋅𝐵.

Proof. (a)⇔(b) If 𝑚=|𝐽| and 𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚,𝑣↦𝐷𝑞_𝐽(𝑣)=𝜕𝑞_𝐽𝜕𝑥₁(𝑣),…,𝜕𝑞_𝐽𝜕𝑥_𝑚(𝑣), (39) is the gradient group homomorphism of 𝑞_𝐽, then Ker𝑞_𝐽=Ker[𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚] and the subgroup Ker𝑞_𝐽 of ℤ^𝑚 is of rank 𝑚−rk𝐺_𝐽 and consists of all integral solutions of the system 2⋅𝐺_𝐽⋅𝑥^tr=0 of linear equations with integral coefficients; see [34, Proposition 2.8]. Then, (a)⇔(b) follows.
The equivalences (a)⇔(c)⇔(d) follow from Proposition 5 (a) and the commutativity of the diagram (36).
(e)⇒(a) Assume that there exist a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑠,𝑠≥3,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} and a ℤ-invertible matrix 𝐵∈Gl(𝑚,ℤ) such that 𝐺_𝐷𝐽=𝐵^tr⋅𝐺⋅𝐵. It follows that the quadratic form 𝑞_𝐷𝐽(𝑥)=𝑥⋅𝐺_𝐷𝐽⋅𝑥^tr is ℤ-congruent to 𝑞_𝐽 and 𝑞_𝐽=𝑞_𝐷𝐽∘ℎ_𝐵. Then, (a) is a consequence of [36, Lemma VII.4.2].
(a)⇒(e) Let Δ_𝐽 be the Euler edge-bipartite graph defined in (33) of 𝐽. By (a) and Lemma 3 (d), Δ_𝐽 is principal and the inflation algorithm defined in [4, 21] applies to Δ_𝐽. Consequently, there exists a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑠,𝑠≥3,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} and a ℤ-invertible matrix 𝐵∈Gl(𝑛,ℤ) defining the congruence Δ_𝐽≈_ℤ𝐷𝐽; that is, the equality 𝐺_𝐷𝐽=𝐵^tr⋅𝐺_Δ𝐽⋅𝐵=𝐵^tr⋅𝐺_𝐽⋅𝐵 holds. Then, in view of Proposition 5, the implication (a)⇒(e) follows from Lemma 3 (d); see also Section 6.

Theorem 10. Let 𝐽 be a finite principal poset, with a numbering {𝑎₁,…,𝑎_𝑚} of elements of 𝐽. Fix a nonzero vector 𝐡_𝐽∈ℤ^𝐽≡ℤ^𝑚 such that Ker𝑞_𝐽=ℤ⋅𝐡_𝐽.(a) There exist a minimal integer 𝐜^̌‌_𝐽≥2  (called the reduced Coxeter number of 𝐽) and a group homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ(called the incidence defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)⋅𝐡_𝐽,𝜕_𝐽Φ_𝐽(𝑣)=𝜕_𝐽(𝑣),∀𝑣∈ℤ^𝐽,𝜕_𝐽𝐡_𝐽=0. (40)(b) Assume that 𝐜^̌‌_𝐽≥1 and 𝜕_𝐽∶ℤ^𝐽→ℤ are as in (a), and one sets 𝐡_𝐽=𝐡_𝐽⋅𝐵^′, 𝐡_𝐽=𝐡_𝐽⋅𝐵^′⋅𝐵⁻¹, where 𝐵^′,𝐵∈𝕄_𝐽(ℤ) are as in Proposition 5.(b1) There exists a group homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ(called the Euler defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)⋅𝐡_𝐽,∀𝑣∈ℤ^𝐽,𝜕_𝐽∘Φ_𝐽=𝜕_𝐽,𝜕_𝐽∘ℎ_𝐵′=𝜕_𝐽,𝜕_𝐽𝐡_𝐽=0. (41) (b2) There exists a group homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ  (called the Tits defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)⋅𝐡_𝐽,∀𝑣∈ℤ^𝐽,𝜕_𝐽=𝜕_𝐽∘ℎ_𝐵=𝜕_𝐽∘ℎ_𝐵⁻¹_′∘ℎ_𝐵,𝜕_𝐽=𝜕_𝐽∘Φ_𝐽,𝜕_𝐽𝐡_𝐽=0. (42)(c) The Coxeter number 𝐜_𝐽 of 𝐽 is infinite, and the incidence defect 𝜕_𝐽∶ℤ^𝐽→ℤ is nonzero.(d) Given 𝑣∈ℤ^𝐽, the order 𝐬_𝑣∶=|𝒪(𝑣)| of the Φ_𝐽-orbit 𝒪(𝑣) is finite if and only if 𝜕_𝐽(𝑣)=0. If 𝐬_𝑣=|𝒪(𝑣)| is finite, then 𝐬_𝑣 divides 𝐜^̌‌_𝐽 and there is a unique integer 𝑚_𝑣 such that 𝑚_𝑣⋅𝐡=𝑣+Φ_𝐽(𝑣)+Φ²_𝐽(𝑣)+⋯+Φ^𝐬𝑣_𝐽⁻¹(𝑣)=𝑣+Φ_𝐽⁻¹(𝑣)+Φ_𝐽⁻²(𝑣)+⋯+Φ^−𝐬𝑣_𝐽⁺¹(𝑣). (43)

Proof. We recall from the proof of Proposition 9 that ℤ⋅𝐡_𝐽=Ker𝑞_𝐽=Ker𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚, (44) where 𝑚=|𝐽|≥2 and 𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚,𝑣↦𝐷𝑞_𝐽(𝑣)=((𝜕𝑞_𝐽/𝜕𝑥₁)(𝑣),…,(𝜕𝑞_𝐽/𝜕𝑥_𝑚)(𝑣)), is the gradient group homomorphism. It follows that ℤ^𝑚/ℤ⋅𝐡_𝐽≅Im𝐷𝑞_𝐽≅ℤ^𝑚−1. Denote by 𝜙∶ℤ^𝑚→ℤ^𝑚−1 the composite quotient epimorphism. Then, the form 𝑞_𝐽 induces the form 𝑞^∼_𝐽∶ℤ^𝑚−1→ℤ such that 𝑞^∼_𝐽(𝜙(𝑥))=𝑞_𝐽(𝑥), for all 𝑥∈ℤ^𝑚. Moreover, the Coxeter transformation Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 induces a group automorphism Φ^∼_𝐽∶ℤ^𝑚−1→ℤ^𝑚−1 such that Φ^∼_𝐽∘𝜙=𝜙∘Φ_𝐽,𝑞^∼Φ_𝐽^∼_𝐽𝑦=𝑞^∼_𝐽𝑦,∀𝑦∈ℤ^𝑚−1. (45) It follows that 𝑞^∼_𝐽 is positive definite and there exists a minimal integer 𝐜^̌‌_𝐽≥1 such that Φ^{∼̌‌𝐜𝐽}_𝐽 is the identity map on ℤ^𝑚−1. Hence, (a) follows, because the equalities 𝜕_𝐽(𝐡_𝐽)=0 and 𝜕_𝐽(Φ_𝐽(𝑣))=𝜕_𝐽(𝑣), for all 𝑣∈ℤ^𝐽, are almost obvious; see [34, Theorem 4.7].
In view of Proposition 5, the statements (b)–(d) are a consequence of (a) and Lemma 8(a). The reader is referred to [34, Theorem 4.7, Corollary 4.15] for more details.

Corollary 11. (a) If 𝐽 is a principal connected poset with at most 13 elements, then its Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of a unit circle 𝒮¹={𝑧∈ℂ;|𝑧|=1}, 1∈𝐬𝐩𝐞𝐜𝐜_𝐽, and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐽 is an 𝑚th root of unity, where 𝑚≤𝐜^̌‌_𝐽 and 𝐜^̌‌_𝐽 is the reduced Coxeter number of 𝐽.
(b) If 𝐼 and 𝐽 are one-peak principal posets with at most 13 elements and 𝐷𝐼, 𝐷𝐽 are the associated Euclidean diagrams, then the following conditions are equivalent:(b1)𝐷𝐼=𝐷𝐽,(b2)𝐬𝐩𝐞𝐜𝐜_𝐼=𝐬𝐩𝐞𝐜𝐜_𝐽,(b3)𝐜^̌‌_𝐼=𝐜^̌‌_𝐽 and |𝐼|=|𝐽|,(b4) the incidence matrix 𝐶_𝐽∈𝕄_𝐽(ℤ) is ℤ-congruent to the incidence matrix 𝐶_𝐼∈𝕄_𝐼(ℤ); that is, there is a ℤ-invertible matrix 𝐵 such that 𝐶_𝐼=𝐵^𝑡𝑟⋅𝐶_𝐽⋅𝐵.

Proof. (a) By Lemma 8, 𝐬𝐩𝐞𝐜𝐜_𝐽⊆𝒮¹ and 1∈𝐬𝐩𝐞𝐜𝐜_𝐽. Assume that 𝐷𝐽 is the associated Euclidean diagram of 𝐽, as in Proposition 9. By a computer search (using the results of [43] and the inflation algorithm given in [4, 21]), we show that cox_𝐽(𝑡)=cox_𝐷𝐽(𝑡),𝐜^̌‌_𝐽=𝐜^̌‌_𝐷𝐽, (46) for any poset 𝐽, with at most 13 elements. Hence, in view of [4, Proposition 2.17], we have cox_𝐽(𝑡)=cox_𝐷𝐽(𝑡)=𝐹_𝐷𝐽(𝑡), (47) where 𝐹_𝐷𝐽(𝑡)=𝑡^𝑛+1+𝑡^𝑛−𝑡^𝑛−1−𝑡^𝑛−2−𝑡³−𝑡²=()+𝑡+1𝑡−1²𝔳²₂(𝑡)𝔳_𝑛−2(𝑡),for𝐷𝐽=𝔻^∼_𝑛,𝑡⁷+𝑡⁶−2𝑡⁴−2𝑡³=()+𝑡+1𝑡−1²𝔳₂(𝑡)𝔳²₃(𝑡),for𝐷𝐽=𝔼^∼₆,𝑡⁸+𝑡⁷−𝑡⁵−2𝑡⁴−𝑡³=()+𝑡+1𝑡−1²𝔳₂(𝑡)𝔳₃(𝑡)𝔳₄(𝑡),for𝐷𝐽=𝔼^∼₇,𝑡⁹+𝑡⁸−𝑡⁶−𝑡⁵−𝑡⁴−𝑡³=()+𝑡+1𝑡−1²𝔳₂(𝑡)𝔳₃(𝑡)𝔳₅(𝑡),for𝐷𝐽=𝔼^∼₈, (48) where 𝔳_𝑚(𝑡)=𝑡^𝑚−1+𝑡^𝑚−2+𝑡^𝑚−3+⋯+𝑡²+𝑡+1. For 𝐷𝐽∈{𝔻^∼₄,𝔻^∼₅}, we have 𝐹_𝐷𝐽(𝑡)=𝑡⁵+𝑡⁴−2𝑡³−2𝑡²+𝑡+1,for𝐷𝐽=𝔻^∼₄,𝑡⁶+𝑡⁵−𝑡⁴−2𝑡³−𝑡²+𝑡+1,for𝐷𝐽=𝔻^∼₅. (49)
Then, (a) follows by applying [38, Lemma XIII.1.3]. Hence, we also easily conclude that the statements (b1)–(b3) are equivalent.
To finish the proof of (b), we note that the equality 𝐶_𝐼=𝐵^tr⋅𝐶_𝐽⋅𝐵 in (b4) implies that the matrices Cox_𝐼 and Cox_𝐽 are conjugate, and, hence, we get 𝐬𝐩𝐞𝐜𝐜_𝐼=𝐬𝐩𝐞𝐜𝐜_𝐽; that is, the implication (b4)⇒(b2) holds. To prove the inverse implication (b2)⇒(b4), we apply the technique used in [18, Section 6]. On this way, given a principal poset 𝐽, with at most 13 elements and the associated Euclidean diagram 𝐷𝐽, we construct (by a computer search) a ℤ-invertible matrix 𝐵_𝐽 such that 𝐺^̌‌_𝐷𝐽=𝐵_𝐽^tr⋅𝐶_𝐽⋅𝐵_𝐽 (compare with [17, 18, 33, 43]). Hence, (b4) follows, and the proof is complete.

If 𝐽 is a principal poset, then the sets ℛ_𝑞𝐽=𝑣∈ℤ^𝑚;𝑞_𝐽(𝑣),ℛ𝑞=1_𝐽=𝑣∈ℤ^𝑚;𝑞_𝐽(𝑣),ℛ=1_𝑞𝐽=𝑣∈ℤ^𝑚;𝑞_𝐽(𝑣)=1 (50) of roots of the unit forms 𝑞_𝐽, 𝑞_𝐽, and 𝑞_𝐽 have the disjoint union decompositions ℛ_𝑞𝐽=𝜕⁻_𝐽ℛ_𝑞𝐽∪𝜕⁺_𝐽ℛ_𝑞𝐽∪𝜕⁰_𝐽ℛ_𝑞𝐽,ℛ𝑞_𝐽=𝜕⁻_𝐽ℛ𝑞_𝐽∪𝜕⁺_𝐽ℛ𝑞_𝐽∪𝜕⁰_𝐽ℛ𝑞_𝐽,ℛ_𝑞𝐽=𝜕⁻_𝐽ℛ_𝑞𝐽∪𝜕⁺_𝐽ℛ_𝑞𝐽∪𝜕⁰_𝐽ℛ_𝑞𝐽, (51) where 𝜕⁻_𝐽ℛ_𝑞𝐽=𝑣∈ℛ_𝑞𝐽;𝜕_𝐽(𝑣),𝜕<0_+𝐽ℛ_𝑞𝐽=𝑣∈ℛ_𝑞𝐽;𝜕_𝐽(𝑣),𝜕>0_0𝐽ℛ_𝑞𝐽=𝑣∈ℛ_𝑞𝐽;𝜕_𝐽(𝑣);𝜕=0_−𝐽ℛ𝑞_𝐽=𝑣∈ℛ𝑞_𝐽;𝜕_𝐽(𝑣),𝜕<0_+𝐽ℛ𝑞_𝐽=𝑣∈ℛ𝑞_𝐽;𝜕_𝐽(𝑣),𝜕>0_0𝐽ℛ𝑞_𝐽=𝑣∈ℛ𝑞_𝐽;𝜕_𝐽(𝑣)=0;𝜕⁻_𝐽ℛ_𝑞𝐽=𝑣∈ℛ_𝑞𝐽;𝜕_𝐽(𝑣)<0,𝜕⁺_𝐽ℛ_𝑞𝐽=𝑣∈ℛ_𝑞𝐽;𝜕_𝐽(𝑣)>0,𝜕⁰_𝐽ℛ_𝑞𝐽=𝑣∈ℛ_𝑞𝐽;𝜕_𝐽(𝑣)=0. (52)

Note that the group isomorphism ℤ^𝐽→ℤ^𝐽, 𝑣↦𝑣∶=−𝑣, restricts to the bijections 𝜕⁻_𝐽ℛ_𝑞𝐽≃₁→₋₁𝜕⁺_𝐽ℛ_𝑞𝐽,𝜕⁻_𝐽ℛ𝑞_𝐽≃₁→₋₁𝜕⁺_𝐽ℛ𝑞_𝐽,𝜕⁻_𝐽ℛ_𝑞𝐽≃₁→₋₁𝜕⁺_𝐽ℛ_𝑞𝐽. (53)

Example 12. We compute the reduced Coxeter number, the Coxeter polynomial, and the Euler defect of the following principal two-peak poset (54) Note that 𝐽 is principal, because 𝑞_𝐽(𝑥)=𝑥²₁+𝑥²₂+𝑥²₃+𝑥²₄+𝑥²₅+𝑥²₆+𝑥²₇−𝑥₁+𝑥₂𝑥₃−𝑥₁+𝑥₅𝑥₄−𝑥₁𝑥₆+𝑥₁−𝑥₂−𝑥₅−𝑥₆𝑥₇=𝑥₁−12𝑥₄−12𝑥₅−12𝑥₆+12𝑥₇²+𝑥₂−12𝑥₃−12𝑥₇²+5𝑥12³−25𝑥₅−45𝑥₆+15𝑥₇²+34−13𝑥₃+𝑥₄−23𝑥₅−13𝑥₆+13𝑥₇²+35𝑥₅−12𝑥₆−12𝑥₇²+14𝑥₆−𝑥₇². (55)
It follows that 𝑞_𝐽 is nonnegative and Ker𝑞_𝐽=ℤ⋅𝐡, where 𝐡=(1,1,1,1,1,1,1); 𝑞_𝐽 is critical in the sense of Ovsienko [24]; see also [38, 44]. Note that the Euler matrix 𝐶_𝐽=𝐶_𝐽⁻¹ of 𝐽 and the inverse of the Coxeter-Euler matrix Cox_𝐽∶=−𝐶_𝐽⁻¹⋅𝐶_𝐽^tr have the forms 𝐶_𝐽=𝐶_𝐽⁻¹=,10−1−10−1101−1000−100100000001−100000010−1000001−10000001Cox_𝐽⁻¹=.−10−1−1−1−1−10−1−1000−11111112101001000010001011100−110−1001 (56)
Moreover, we have 𝐺^∼_𝔼6=𝐵^tr⋅𝐺_𝐽⋅𝐵, and the matrix 𝐴∶=𝐵^tr⋅𝐶_𝐽⋅𝐵 is a morsification of the Euclidean diagram 𝔼^∼₆ (see [34, 40]), where ,.𝐵=−20100000000001−1000010−10001000001000−11000001000000𝐴=11−21110−21000002−110−1−10−1001000−100−1100−100001000000−11 (57) Hence, in view of [2, Proposition 2.8], we get the following:(i) the Euclidean type 𝐷𝐽 of 𝐽 is the diagram 𝔼^∼₆, and the Coxeter polynomial of the poset 𝐽 has the form cox_𝐽(𝑡)=𝑡⁷−𝑡⁵−𝑡²+1=cox_𝐴(𝑡); (58) that is, cox_𝐽(𝑡) is the Coxeter polynomial 𝐹_Δ⁽²⁾(𝑡) (21), of the Euclidean diagram Δ=𝔸^∼₆ (with a particular numbering of vertices), and is the Coxeter polynomial cox_𝐴(𝑡) of the morsification 𝐴∈𝕄₇(ℤ) of the diagram 𝔼^∼₆,(ii) the Coxeter number 𝐜_𝐽 is infinite and the reduced Coxeter number 𝐜^̌‌_𝐽 equals 10,(iii) the Euler defect has the form 𝜕_𝐽(𝑥)=3(𝑥₁+𝑥₂−𝑥₃−𝑥₇), (iv) the Φ_𝐽-orbit of any vector of defect zero in 𝜕⁰_𝐽ℛ𝑞_𝐽 is of length 2 or of length 5. It is shown in [1, Remark 4.5] and [34, Example 5.14] that they lie on one sand-glass tube 𝒯_1,2 of rank 2 and on six sand-glass tubes of rank five.

4. Nonnegative Posets of Positive Corank

In the study of nonnegative posets, the following extensions of [34, Definition 2.2] are of importance.

Definition 13. Assume that 𝑚≥2, 𝑟≥0, and 𝑞∶ℤ^𝑚→ℤ is a unit quadratic form.(a)The form 𝑞 is defined to be nonnegative of corank  𝑟≥0, if 𝑞 is nonnegative and the ℚ-rank rk_ℚ𝐺_𝑞 of the rational Gram matrix 𝐺_𝑞∈𝕄_𝑚(ℚ) equals rk_ℚ𝐺_𝑞=𝑚−𝑟.(b)The form 𝑞 is defined to be nonnegative critical of corank  𝑟≥1, if 𝑞 is nonnegative of corank 𝑟≥1 and each of the nonnegative quadratic forms 𝑞⁽¹⁾,…,𝑞^(𝑚)∶ℤ^𝑚−1→ℤ is of corank at most 𝑟−1≥0, where 𝑞^(𝑗)𝑥₁,…,𝑥_𝑗−1,𝑥_𝑗+1,…,𝑥_𝑚=𝑞𝑥₁,…,𝑥_𝑗−1,0,𝑥_𝑗+1,…,𝑥_𝑚. (59)

Lemma 14. Assume that 𝑚≥2, 𝑟≥0, and 𝑞∶ℤ^𝑚→ℤ is an integral quadratic form.(a)𝑞 is nonnegative of corank 𝑟≥0 if and only if 𝑞 is nonnegative and the subgroup Ker𝑞 of the abelian group ℤ^𝑚 is free of rank 𝑟.(b)𝑞 is nonnegative of corank 𝑟=0 if and only if 𝑞 is positive, and 𝑞 is nonnegative of corank one if and only if 𝑞 is principal.(c)𝑞 is nonnegative critical of corank 𝑟≥1 if and only if 𝑞 is nonnegative and, for any 𝑗∈{1,…,𝑚}, the abelian subgroup ℤ^𝑚,𝑗∩Ker𝑞 of ℤ^𝑚 is free of rank at most 𝑟−1, where ℤ^𝑚,𝑗∶=ℤ^{𝑚−𝑗−1}×{0}×ℤ^𝑗−1=𝑣=𝑣₁,…,𝑣_𝑚∈ℤ^𝑚;𝑣_𝑗=0⊆ℤ^𝑚 (60) is viewed as a subgroup of ℤ^𝑚.(d)𝑞 is nonnegative critical of corank 𝑟=1 if and only if 𝑞 is 𝑃-critical in the sense of [34, Definition 2.2] and [44].

Proof. The proof of (a) follows by applying the arguments used in the proof of the equivalence (a)⇔(b) in Proposition 9. The statements (b) and (c) follow from (a).
(c) First, we note that the quadratic forms 𝑞⁽¹⁾,…,𝑞^(𝑚)∶ℤ^𝑚−1→ℤ are nonnegative, if 𝑞 is nonnegative. Then, (c) is a consequence of the group isomorphism Ker𝑞^≃(𝑗)−→ℤ^𝑚,𝑗𝑤∩Ker𝑞,𝑤↦^(𝑗)∶=𝑤₁,…,𝑤_𝑗−1,0,𝑤_𝑗+1,…,𝑤_𝑚. (61) Since (d) is a consequence of (c), the proof is complete.

Definition 15. Assume that 𝐽 is a connected poset and 𝑞_𝐽,𝑞_𝐽∶ℤ^𝐽→ℤ are its incidence and Tits quadratic forms (6), respectively.(a)𝐽 is defined to be nonnegative of corank  𝑟≥0 if its incidence quadratic form 𝑞_𝐽∶ℤ^𝐽→ℤ (resp., one of the forms 𝑞_𝐽 and 𝑞_𝐽) is nonnegative and the free abelian subgroup Ker𝑞_𝐽 of ℤ^𝐽 is of ℤ-rank 𝑟 (resp., Ker𝑞_𝐽≅Ker𝑞_𝐽≅Ker𝑞_𝐽 is of ℤ-rank 𝑟); see (36).(b)𝐽 is defined to be nonprincipal critical if the incidence quadratic form 𝑞_𝐽∶ℤ^𝐽→ℤ is nonnegative and not positive, 𝐽 is not principal, and the quadratic form 𝑞_𝐽′∶ℤ^𝐽′→ℤ is principal or positive, for every proper subposet 𝐽^′ of 𝐽.(c)A one-peak poset 𝐼, with max𝐼={∗}, is defined to be nonprincipal Tits-critical if the Tits quadratic form 𝑞_𝐼∶ℤ^𝐼→ℤ is nonnegative and not positive, 𝐼 is not principal, and the Tits quadratic form 𝑞_𝐼′∶ℤ^𝐼′→ℤ is principal or positive, for every proper subposet 𝐼^′ of 𝐼 containing the peak ∗. We call a nonprincipal Tits-critical poset 𝐼  exceptional, if the subposet 𝑇=𝐼⧵{∗} is nonprincipal Tits-critical; see [33, 34].(d)A poset 𝐽 is defined to be 𝑃-hypercritical if 𝐽 is not nonnegative and each of its proper subposet is nonnegative; see [34, Definition 2.2].

Remark 16. Assume that 𝑇 is a poset and 𝑇^∗=𝑇∪{∗} is its one-peak enlargement.(a)If 𝑇^∗ is 𝑃-hypercritical, then 𝑇 is 𝑁𝑃-critical in the sense of [14], but not conversely.(b)By [43], many of the 𝑁𝑃-critical posets 𝑇 listed in [14, Table 2] are of corank at most two.(c)A Coxeter spectral classification of one-peak positive (resp., almost Tits 𝑃-critical) posets is given in [17, 18] (resp., in [33]).
We frequently use the following important characterisation.

Theorem 17. Assume that 𝐽 is a connected poset and 𝑞_𝐽,𝑞_𝐽∶ℤ^𝐽→ℤ are the incidence and the Tits quadratic forms of 𝐽 (7), respectively.(a) If 𝐽 is nonnegative of corank two, then 𝐽 contains at least 6 elements, and |𝐽|=6 if and only if 𝐽 is the garland (62) and Ker𝑞_𝒢3∶=ℤ⋅𝐡⁽¹⁾⊕ℤ⋅𝐡⁽²⁾, where 𝐡⁽¹⁾=(1,1,−1,−1,0,0) and 𝐡⁽²⁾=(1,1,0,0,−1,−1). The garland 𝒢₃ is nonprincipal critical.(b) The following four conditions are equivalent.(b1) The poset 𝐽 is nonprincipal critical.(b2)|𝐽|≥6 and the form 𝑞_𝐽∶ℤ^𝐽→ℤ is nonnegative critical of corank two.(b3)|𝐽|≥6 and 𝑞_𝐽∶ℤ^𝐽→ℤ is nonnegative, the group Ker𝑞_𝐽 is of ℤ-rank two, and for any 𝑗∈𝐽, the subposet 𝐽^(𝑗)∶=𝐽⧵{𝑗} of 𝐽 is principal or positive.(b4)|𝐽|≥6 and 𝑞_𝐽∶ℤ^𝐽→ℤ is nonnegative, and the group Ker𝑞_𝐽 has a ℤ-basis 𝐡,𝐡^′ such that there is no 𝑗∈𝐽, with ℎ_𝑗=ℎ^′_𝑗=0.(c) Let 𝐼 be a one-peak poset 𝐼, with max𝐼={∗}. The following three conditions are equivalent.(c1)𝐼 is nonprincipal Tits-critical.(c2)|𝐼|≥7, the Tits form 𝑞_𝐼∶ℤ^𝐼→ℤ is nonnegative, the group Ker𝑞_𝐼 is of ℤ-rank two, and for any 𝑗∈𝐼⧵{∗}, the one-peak subposet 𝐼^(𝑗)∶=𝐼⧵{𝑗} of 𝐼 is principal or positive.(c3)|𝐼|≥7 and 𝑞_𝐼∶ℤ^𝐼→ℤ is nonnegative, and the group Ker𝑞_𝐼 has a ℤ-basis 𝐡,𝐡^′ such that there is no 𝑗∈𝐼⧵{∗}, with ℎ_𝑗=ℎ^′_𝑗=0. (d) A nonprincipal Tits-critical one-peak poset 𝐼, with max𝐼={∗} and |𝐼|=7, is exceptional if and only if 𝐼 is the one-peak garland (63)and Ker𝑞_𝒢^∗₃∶=ℤ⋅𝐡⁽¹⁾⊕ℤ⋅𝐡⁽²⁾, where 𝐡⁽¹⁾=(1,1,−1,−1,0,0,0), 𝐡⁽²⁾=(1,1,0,0,−1,−1,0).

Proof. (a) It is easy to check that any poset 𝐽 with at most 5 elements is either positive or principal. Moreover, if 𝐽 is nonnegative of corank two and |𝐽|=6, then 𝐽 is the garland 𝒢₃. Since 𝑞_𝒢3(𝑥)=𝑥²₁+𝑥²₂+𝑥²₃+𝑥²₄+𝑥²₅+𝑥²₆+𝑥₁+𝑥₂𝑥₃+𝑥₄+𝑥₅+𝑥₆+𝑥₃+𝑥₄𝑥₅+𝑥₆, (64) the Lagrange’s algorithm yields 𝑞_𝒢3(𝑥)=𝑥₁+12𝑥₃+12𝑥₄+12𝑥₅+12𝑥₆²+𝑥₂+12𝑥₃+12𝑥₄+12𝑥₅+12𝑥₆²+12𝑥₃−𝑥₄²+12𝑥₅−𝑥₆². (65) It follows that 𝑞_𝒢3∶ℤ⁶→ℤ is nonnegative and its kernel is a rank-two free abelian group of the form shown in (a). Hence, (a) follows.
(b) We show by a computer search that there is no nonprincipal critical poset 𝐽 such that |𝐽|≤5. Then, in view of Lemma 14, the equivalences (b1)⇔(b2)⇔(b3)⇔(b4) easily follow.
(c) We show by a computer search that there is no one-peak nonprincipal Tits-critical poset 𝐼 such that |𝐼|≤6. Then, in view of Lemma 14, the equivalences (c1)⇔(c2)⇔(c3) easily follow.
(d) Note that 𝑞_𝒢^∗₃(𝑥)=𝑥²₁+𝑥²₂+𝑥²₃+𝑥²₄+𝑥²₅+𝑥²₆+𝑥²₇+𝑥₁+𝑥₂𝑥₃+𝑥₄+𝑥₅+𝑥₆+𝑥₃+𝑥₄𝑥₅+𝑥₆−𝑥₁+𝑥₂+𝑥₃+𝑥₄+𝑥₅+𝑥₆𝑥₇, (66) and the Lagrange's algorithm yields 𝑞_𝒢^∗₃(𝑥)=𝑥₁+12𝑥₃+12𝑥₄+12𝑥₅+12𝑥₆−12𝑥₇²+𝑥₂+12𝑥₃+12𝑥₄+12𝑥₅+12𝑥₆−12𝑥₇²+12𝑥₃−𝑥₄²+12𝑥₅−𝑥₆²+12𝑥²₇. (67) It follows that 𝑞_𝒢^∗₃∶ℤ⁷→ℤ is nonnegative and its kernel is a rank-two free abelian group of the form shown in (d). Hence, the one-peak garland 𝒢^∗₃ is nonprincipal Tits-critical and exceptional. On the other hand, one shows by a computer search that 𝒢^∗₃ is the only one-peak poset that is nonprincipal Tits-critical and exceptional. This finishes the proof.

Following [34, Section 4], we will study nonnegative posets 𝐽 of corank 𝑟≥2 by means of the spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽, the reduced Coxeter number 𝐜^̌‌_𝐽, and the rank 𝑟≥2 defects 𝜕_𝐽=𝜕_𝐽⁽¹⁾,…,𝜕_𝐽^(𝑟),𝜕_𝐽=𝜕_𝐽⁽¹⁾,…,𝜕_𝐽^(𝑟),𝜕_𝐽=𝜕_𝐽⁽¹⁾,…,𝜕_𝐽^(𝑟)∶ℤ^𝐽→ℤ^𝑟 (68) defined in the following extension of Theorem 10.

Theorem 18. Let 𝐽 be a finite nonnegative poset of corank 𝑟≥2, and let 𝑚=|𝐽|≥2. One fixes nonzero vectors 𝐡_𝐽⁽¹⁾,…,𝐡_𝐽^(𝑟)∈ℤ^𝐽 such that Ker𝑞_𝐽=ℤ⋅𝐡_𝐽⁽¹⁾⊕⋯⊕ℤ⋅𝐡_𝐽^(𝑟)≅ℤ^𝑟, and one sets 𝐡_𝐽=(𝐡_𝐽⁽¹⁾,…,𝐡_𝐽^(𝑟)).(a) There exist a minimal integer 𝐜^̌‌_𝐽≥1 (called the reduced Coxeter number of 𝐽) and a group homomorphism 𝜕_𝐽=(𝜕_𝐽⁽¹⁾,…,𝜕_𝐽^(𝑟))∶ℤ^𝐽→ℤ^𝑟≅Ker𝑞_𝐽(called the incidence defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)•𝐡_𝐽=𝑣+𝜕_𝐽⁽¹⁾(𝑣)⋅𝐡_𝐽⁽¹⁾+⋯+𝜕_𝐽^(𝑟)(𝑣)⋅𝐡_𝐽^(𝑟),𝜕_𝐽Φ_𝐽(𝑣)=𝜕_𝐽(𝑣),∀𝑣∈ℤ^𝐽, (69) and 𝜕_𝐽(𝐡)=0, for all 𝐡∈Ker𝑞_𝐽, where one sets 𝜕_𝐽(𝑣)=𝜕_𝐽⁽¹⁾(𝑣),…,𝜕_𝐽^(𝑟)(𝑣),𝜕_𝐽(𝑣)•𝐡_𝐽∶=𝜕_𝐽⁽¹⁾(𝑣)⋅𝐡_𝐽⁽¹⁾+⋯+𝜕_𝐽^(𝑟)(𝑣)⋅𝐡_𝐽^(𝑟). (70)(b) Assume that 𝐜^̌‌_𝐽≥1 and 𝜕_𝐽∶ℤ^𝐽→ℤ^𝑟 are as in (a), and one sets 𝐡_𝐽⁽¹⁾=𝐡_𝐽⁽¹⁾⋅𝐵^′,…,𝐡_𝐽^(𝑟)=𝐡_𝐽^(𝑟)⋅𝐵^′,𝐡_𝐽⁽¹⁾=𝐡_𝐽⁽¹⁾⋅𝐵^′⋅𝐵⁻¹,…,𝐡_𝐽^(𝑟)=𝐡_𝐽^(𝑟)⋅𝐵^′⋅𝐵⁻¹, (71) where 𝐵^′,𝐵∈𝕄_𝐽(ℤ) are as in Proposition 5.(b1) There exists a group homomorphism 𝜕_𝐽=(𝜕_𝐽⁽¹⁾,…,𝜕_𝐽^(𝑟))∶ℤ^𝐽→ℤ^𝑟≅Ker𝑞_𝐽(called the Euler defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)•𝐡_𝐽=𝑣+𝜕_𝐽⁽¹⁾(𝑣)⋅𝐡_𝐽⁽¹⁾+⋯+𝜕_𝐽^(𝑟)(𝑣)⋅𝐡_𝐽^(𝑟),∀𝑣∈ℤ^𝐽, (72) 𝜕_𝐽∘Φ_𝐽=𝜕_𝐽,𝜕_𝐽=𝜕_𝐽∘ℎ_𝐵′, and 𝜕_𝐽(𝐡)=0,  for all 𝐡∈Ker𝑞_𝐽, where one sets 𝐡_𝐽=𝐡_𝐽⁽¹⁾,…,𝐡_𝐽^(𝑟),𝜕_𝐽(𝑣)•𝐡_𝐽∶=𝜕_𝐽⁽¹⁾(𝑣)⋅𝐡_𝐽⁽¹⁾+⋯+𝜕_𝐽^(𝑟)(𝑣)⋅𝐡_𝐽^(𝑟). (73)(b2) There exists a group homomorphism 𝜕_𝐽=(𝜕_𝐽⁽¹⁾,…,𝜕_𝐽^(𝑟))∶ℤ^𝐽→ℤ^𝑟≅Ker𝑞_𝐽(called the Tits defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)•𝐡_𝐽=𝑣+𝜕_𝐽⁽¹⁾(𝑣)⋅𝐡_𝐽⁽¹⁾+⋯+𝜕_𝐽^(𝑟)(𝑣)⋅𝐡_𝐽^(𝑟),∀𝑣∈ℤ^𝐽, (74) 𝜕_𝐽∘Φ_𝐽=𝜕_𝐽, 𝜕_𝐽=𝜕_𝐽∘ℎ_𝐵=𝜕_𝐽∘ℎ_𝐵⁻¹_′∘ℎ_𝐵, and 𝜕_𝐽(𝐡)=0,  for all 𝐡∈Ker𝑞_𝐽, where one sets 𝐡_𝐽=𝐡_𝐽⁽¹⁾,…,𝐡_𝐽^(𝑟),𝜕_𝐽(𝑣)•𝐡_𝐽∶=𝜕_𝐽⁽¹⁾(𝑣)⋅𝐡_𝐽⁽¹⁾+⋯+𝜕_𝐽^(𝑟)(𝑣)⋅𝐡_𝐽^(𝑟). (75)(c) The Coxeter number 𝐜_𝐽 of 𝐽 is finite if and only if the incidence defect 𝜕_𝐽∶ℤ^𝐽→ℤ^𝑟 is zero. In this case, 𝐜^̌‌_𝐽=𝐜_𝐽.(d) Given 𝑣∈ℤ^𝑚≡ℤ^𝐽, the order 𝐬_𝑣∶=|𝒪(𝑣)| of the Φ_𝐽-orbit 𝒪(𝑣) is finite if and only if𝜕_𝐽(𝑣)=0. If𝐬_𝑣=|𝒪(𝑣)| is finite, then 𝐬_𝑣 divides 𝐜^̌‌_𝐽 and there is a unique integer 𝑚_𝑣 such that 𝑚_𝑣⋅𝐡=𝑣+Φ_𝐽(𝑣)+Φ²_𝐽(𝑣)+⋯+Φ^𝐬𝑣_𝐽⁻¹(𝑣)=𝑣+Φ_𝐽⁻¹(𝑣)+Φ_𝐽⁻²(𝑣)+⋯+Φ^−𝐬𝑣_𝐽⁺¹(𝑣). (76)(e) The statement (d) holds with Φ_𝐽 and Φ_𝐽  (resp., Φ_𝐽) interchanged.

Proof. For simplicity of presentation, we assume that 𝑟=2. We recall from the proof of Proposition 9 that ℤ⋅𝐡_𝐽⁽¹⁾⊕ℤ⋅𝐡_𝐽⁽²⁾=Ker𝑞_𝐽=Ker[𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚], where 𝑚=|𝐽|≥2 and 𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚,𝑣↦𝐷𝑞_𝐽(𝑣)=𝜕𝑞_𝐽𝜕𝑥₁(𝑣),…,𝜕𝑞_𝐽𝜕𝑥_𝑚(𝑣), (77) is the gradient group homomorphism. It follows that ℤ^𝑚ℤ⋅𝐡_𝐽⁽¹⁾⊕ℤ⋅𝐡_𝐽⁽²⁾=ℤ^𝑚Ker𝑞_𝐽≅Im𝐷𝑞_𝐽≅ℤ^𝑚−2. (78) Denote by 𝜙∶ℤ^𝑚→ℤ^𝑚−2 the composite quotient epimorphism. Then, the form 𝑞_𝐽 induces the form 𝑞^∼_𝐽∶ℤ^𝑚−2→ℤ such that 𝑞^∼_𝐽(𝜙(𝑥))=𝑞_𝐽(𝑥), for all 𝑥∈ℤ^𝑚. Moreover, the Coxeter transformation Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 induces a group automorphism Φ^∼_𝐽∶ℤ^𝑚−2→ℤ^𝑚−2 such that Φ^∼_𝐽∘𝜙=𝜙∘Φ_𝐽,𝑞^∼Φ_𝐽^∼_𝐽𝑦=𝑞^∼_𝐽𝑦, (79) for all 𝑦∈ℤ^𝑚−2. It follows that 𝑞^∼_𝐽 is positive definite, and there exists a minimal integer 𝐜^̌‌_𝐽≥1 such that Φ^{∼̌‌𝐜𝐽}_𝐽 is the identity map on ℤ^𝑚−2. Hence, given 𝑣∈ℤ^𝑚, the element Φ^{̌‌𝐜𝐽}_𝐽(𝑣)−𝑣 lies in the kernel of 𝑞_𝐽; that is, it has the form Φ^{̌‌𝐜𝐽}_𝐽(𝑣)−𝑣=𝜕_𝐽⁽¹⁾(𝑣)⋅𝐡_𝐽⁽¹⁾+𝜕_𝐽⁽²⁾(𝑣)⋅𝐡_𝐽⁽²⁾, (80) where 𝜕_𝐽⁽¹⁾(𝑣),𝜕_𝐽⁽²⁾(𝑣) are integers uniquely determined by 𝑣. Since Φ_𝐽 is a group homomorphism, then 𝜕_𝐽⁽¹⁾𝑣+𝑣^′=𝜕_𝐽⁽¹⁾(𝑣)+𝜕_𝐽⁽¹⁾𝑣^′,𝜕_𝐽⁽²⁾𝑣+𝑣^′=𝜕_𝐽⁽²⁾(𝑣)+𝜕_𝐽⁽²⁾𝑣^′; (81) that is, we have defined a pair of group homomorphisms 𝜕_𝐽⁽¹⁾,𝜕_𝐽⁽²⁾∶ℤ^𝐽→ℤ; hence, 𝜕_𝐽=(𝜕_𝐽⁽¹⁾,𝜕_𝐽⁽²⁾)∶ℤ^𝐽→ℤ² is a group homomorphism. It is easy to see that 𝜕_𝐽 has the properties required in (a), and (a) follows.
In view of Proposition 5, the statements (b)–(e) are a consequence of (a). The reader is referred to [34, Theorem 4.17] for more details and a generalization.

Corollary 19. Assume that 𝐽 is a finite nonnegative poset of corank 𝑟≥2.(a)The Coxeter number 𝐜_𝐽 of 𝐽 is infinite if and only if the defect homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ^𝑟 is nonzero, or, equivalently, if and only if the Φ_𝐽-orbit 𝒪(𝑒_𝑗) of some basis vector 𝑒_𝑗∈ℤ^𝐽 is infinite.(b)The Coxeter transformation Φ_𝐽 is weakly periodic in the sense of Sato [42]; that is, Φ^𝑠_𝐽−id is nilpotent, for some 𝑠≥1.

Proof. The statement (a) follows immediately from Theorem 18. To prove (b), we check that (Φ^{̌‌𝐜𝐽}_𝐽−𝑖𝑑)²=0.

Remark 20. (a) It was shown in [34, Example 5.18] that, for the one-peak garland 𝐼=𝒢^∗₃ of Theorem 17(d), we have(i)𝜕_𝐼=𝜕_𝐼=𝜕_𝐼=0 and 𝐜_𝐼=𝐜^̌‌_𝐼=4,(ii)the set ℛ𝑞_𝐼 of Tits roots of 𝐼 lies on 22 sand-glass tubes; six of them are of rank two, and each of the remaining fourteen tubes is of rank four; see [34, pp. 459–461] for details.
(b) By Lemma 8(a), the Coxeter number 𝐜_𝐽 is infinite, for every principal poset 𝐽.
(c) By Theorem 17, there is no nonnegative connected poset 𝐽 of corank 2, with |𝐽|≤5. Moreover, a minimal such a poset is the garland (82)
(d) We show in [43] that most of the nonnegative connected posets 𝐽 of corank 2, with at most 15 elements, are of zero defect. We also show there that a smallest nonnegative connected poset 𝐽 with nonzero defect has 8 elements and is one of the following two posets: (83) It is easy to check that(i)𝐜^̌‌_𝐽′=𝐜^̌‌_𝐽′′=2,(ii)cox_𝐽′(𝑡)=cox_𝐽′′(𝑡)=𝑡⁸−4𝑡⁶+6𝑡⁴−4𝑡²+1,(iii) the coordinates of the Tits defect 𝜕_𝐽′=(𝜕_𝐽⁽¹⁾_′,𝜕_𝐽⁽²⁾_′)∶ℤ⁸→ℤ² of 𝐽^′, with respect to the ℤ-basis 𝐡_𝐽⁽¹⁾_′=(),𝐡2,0,−1,−1,1,1,2,0_𝐽(2)′=(0,2,1,1,−1,−1,0,2) (84) of Ker𝑞_𝐽′, are given by the formulae 𝜕_𝐽⁽¹⁾_′(𝑥)=𝑥₂−12𝑥₃−12𝑥₄+12𝑥₅+12𝑥₆−𝑥₇,𝜕_𝐽⁽²⁾_′(𝑥)=𝑥₁+12𝑥₃+12𝑥₄−12𝑥₅−12𝑥₆−𝑥₈, (85)(iv) the coordinates of the Tits defect 𝜕_𝐽′′=(𝜕_𝐽⁽¹⁾_′′,𝜕_𝐽⁽²⁾_′′)∶ℤ⁸→ℤ² of 𝐽^′′, with respect to the ℤ-basis 𝐡_𝐽⁽¹⁾_′′=(),𝐡0,−1,−1,0,1,1,0,0_{𝐽(2)′′}=(1,0,0,1,0,0,1,1) (86)
of Ker𝑞_𝐽′′, are given by 𝜕_𝐽⁽¹⁾_′(𝑥)=𝑥₁+𝑥₂+𝑥₃+𝑥₄+𝑥₅+𝑥₆−𝑥₇−𝑥₈,𝜕_𝐽⁽²⁾_′′(𝑥)=2𝑥₁+𝑥₂+𝑥₃+2𝑥₄+𝑥₅+𝑥₆−2𝑥₇−2𝑥₈. (87)

5. An Example

In this section, we illustrate the results of Section 3 by an example of a principal one-peak poset 𝐼 of the Euclidean type 𝐷𝐼=𝔻^∼₄. We give a description of the set ℛ𝑞_𝐼 of roots of 𝑞_𝐼 and the mesh translation quiver Γ(ℛ𝑞_𝐼,Φ_𝐼) together with the decomposition (see (51)) Γℛ𝑞_𝐼,Φ_𝐼=Γ𝜕⁻_𝐼ℛ𝑞_𝐼,Φ_𝐼∪Γ𝜕⁺_𝐼ℛ𝑞_𝐼,Φ_𝐼∪Γ𝜕⁰_𝐼ℛ_𝑞𝐼,Φ_𝐼. (88) Let 𝐼 be the one-peak garland (89) The incidence matrix 𝐶_𝐼, the Tits matrix 𝐶_𝐼, and the Coxeter-Tits matrix Cox_𝐼=−𝐶_𝐼⋅𝐶_𝐼^−tr of 𝐼 are the following: 𝐶_𝐼=,𝐶1011101111001010001100001_𝐼=1000−10100−11110−11101−100001,Cox_𝐼=.01001100010001100101−1−111−1 (90) The Coxeter polynomial cox_𝐼(𝑡), the Tits quadratic form 𝑞_𝐼∶ℤ⁵→ℤ, and the Coxeter-Tits transformation Φ_𝐼∶ℤ⁵→ℤ⁵ of 𝐼 are cox_𝐼(𝑡)=𝑡⁵+𝑡⁴−2𝑡³−2𝑡²+𝑡+1=𝐹^∼_𝔻4(𝑡),𝑞_𝐼(𝑥)=𝑥⋅𝐶_𝐼⋅𝑥^tr=𝑥²₁+𝑥²₂+𝑥²₃+𝑥²₄+𝑥²₅+𝑥₁+𝑥₂𝑥₃+𝑥₄−𝑥₁+𝑥₂+𝑥₃+𝑥₄𝑥₅,Φ_𝐼(𝑥)=𝑥⋅Cox_𝐼=𝑥₂−𝑥₅,𝑥₁−𝑥₅,𝑥₄+𝑥₅,𝑥₃+𝑥₅,𝑥₁+𝑥₂+𝑥₃+𝑥₄−𝑥₅, (91) for 𝑥=(𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅)∈ℤ^𝐼≡ℤ⁵. Note that the Φ_𝐼-orbit 𝒪(𝑒₅) of 𝑒₅ consists of two vectors 𝑒₅ and Φ_𝐼(𝑒₅)=(−1,−1,1,1,−1)=Φ_𝐼⁻¹(𝑒₅). Since 𝑞_𝐼(𝑥)=𝑥₁+12𝑥₃+12𝑥₄−12𝑥₅²+𝑥₂+12𝑥₃+12𝑥₄−12𝑥₅²+12𝑥₃−𝑥₄²+12𝑥²₅, (92) then the form 𝑞_𝐼∶ℤ⁵→ℤ is positive semidefinite, is not positive definite, Ker𝑞_𝐼=ℤ⋅𝐡_𝐼,where𝐡_𝐼=(1,1,−1,−1,0), (93) and Φ_𝐼(𝐡_𝐼)=𝐡_𝐼. This means that 𝑞_𝐼 is principal, but not 𝑃-critical; see [44]. One easily shows that the reduced Coxeter number of 𝐼 equals 𝐜^̌‌_𝐼=2 and the Tits defect 𝜕_𝐼∶ℤ⁵→ℤ of 𝐼 is given by 𝜕_𝐼(𝑥)=−(𝑥₁+𝑥₂+𝑥₃+𝑥₄), because Φ_𝐼≠𝑖𝑑 and Φ²_𝐼(𝑣)=𝑣+𝜕_𝐼(𝑣)⋅𝐡_𝐼, for any 𝑣∈ℤ⁵. The set ℛ𝑞_𝐼 of roots of 𝑞_𝐼 has the disjoint union decomposition (see (51)) ℛ𝑞_𝐼=𝜕⁻_𝐼ℛ𝑞_𝐼∪𝜕⁺_𝐼ℛ𝑞_𝐼∪𝜕⁰_𝐼ℛ𝑞_𝐼, (94) and 𝜕⁻_𝐼ℛ𝑞_𝐼, 𝜕⁺_𝐼ℛ𝑞_𝐼, 𝜕⁰_𝐼ℛ𝑞_𝐼 are Φ_𝐼-invariant subsets of ℛ𝑞_𝐼. Obviously, the Φ_𝐼-orbit 𝒪(𝑣) of any 𝑣∈𝜕⁰_𝐼ℛ𝑞_𝐼 is of length two, whereas the Φ_𝐼-orbit 𝒪(𝑤) of any vector 𝑤∈𝜕⁻_𝐼ℛ𝑞_𝐼∪𝜕⁺_𝐼ℛ𝑞_𝐼 is infinite. By (92), a vector 𝑣=(𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝑣₅)∈ℤ⁵ is a root of 𝑞_𝐼∶ℤ⁵→ℤ if and only if (2𝑣₁+𝑣₃+𝑣₄−𝑣₅)²+(2𝑣₂+𝑣₃+𝑣₄−𝑣₅)²+2(𝑣₃−𝑣₄)²+2𝑣²₅=4. Hence, looking at all possible decompositions 4=𝑎²₁+𝑎²₂+2𝑎²₃+2𝑎²₄, with 𝑎₁,𝑎₂,𝑎₃,,𝑎₄,∈ℤ, we show that 𝑣=(𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝑣₅)∈ℤ⁵ is a root of 𝑞_𝐼∶ℤ⁵→ℤ if and only if 𝑣 or 𝑣∶=−𝑣 is one of the vectors listed in Table 1 or in Table 2.

Table 1 

Table 2 

(1) The Φ_𝐼-orbits in 𝒫_𝐼∶=𝜕⁻_𝐼ℛ𝑞_𝐼. Since 𝜕_𝐼(𝑢)<0, if 𝑢∈{𝑒₁,𝑒₂,𝑒₃,𝑒₄} or 𝑢 is the vector 𝐩₁₂=(1,1,0,0,1), then the Φ_𝐼-orbits of the vectors 𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝐩₁₂ lie in 𝒫_𝐼∶=𝜕⁻_𝐼ℛ𝑞_𝐼, because 𝒫_𝐼 is a Φ_𝐼-invariant subset of ℛ𝑞_𝐼. It is easy to see that the Φ_𝐼-orbits consist of the vectors listed in Table 1.

Throughout this section, we freely use the Φ_𝐼-mesh terminology and notation introduced in [2, 34, 40].

(2)  Φ_𝐼-mesh quiver Γ(𝒫_𝐼,Φ_𝐼)=Γ(𝜕⁻_𝐼ℛ𝑞_𝐼,Φ_𝐼). It follows from our earlier remarks that the set 𝒫_𝐼∶=𝜕⁻_𝐼ℛ𝑞_𝐼 of the negative defect roots of 𝑞_𝐼 splits into the five Φ_𝐼-orbits 𝒪(𝑒₁), 𝒪(𝑒₂), 𝒪(𝑒₃), 𝒪(𝑒₄), 𝒪(𝐩₁₂). By applying the mesh toroidal algorithm defined in [2, 34], one constructs the following infinite Φ_𝐼-mesh translation quiver of the negative defect roots of 𝑞_𝐼; see Figure 2, where we set 𝑎∶=−𝑎 for any positive integer 𝑎≥1.

Figure 2 

Figure 3 

Sand-glass tube of rank (2,1).

(3)   Φ_𝐼-mesh quiver Γ(𝒬_𝐼,Φ_𝐼)=Γ(𝜕⁺_𝐼ℛ𝑞_𝐼,Φ_𝐼). Since the group isomorphism ℤ^𝐼→ℤ^𝐼, 𝑣↦−𝑣, carries roots to roots, Φ_𝐼-meshes to Φ_𝐼-meshes, and Φ_𝐼-orbits to Φ_𝐼-orbits, then it defines the bijections 𝜕⁻_𝐼ℛ𝑞_𝐼→𝜕⁺_𝐼ℛ𝑞_𝐼 and 𝜕⁰_𝐼ℛ𝑞_𝐼→𝜕⁰_𝐼ℛ𝑞_𝐼, because 𝜕_𝐼(−𝑣)=−𝜕_𝐼(𝑣). It follows that the set 𝒬_𝐼∶=𝜕⁺_𝐼ℛ𝑞_𝐼 of the positive defect roots of 𝑞_𝐼 splits into the five Φ_𝐼-orbits 𝒪(𝑒₁̂), 𝒪(𝑒₂̂), 𝒪(𝑒₃̂), 𝒪(𝑒₄̂), 𝒪(𝐩₁₂), and one constructs the infinite Φ_𝐼-mesh translation quiver Γ𝒬_𝐼,Φ_𝐼=Γ𝜕⁺_𝐼ℛ𝑞_𝐼,Φ_𝐼 (95) of the positive defect roots of 𝑞_𝐼 by interchanging any vector 𝑣 in Γ(𝒫_𝐼,Φ_𝐼)=Γ(𝜕⁻_𝐼ℛ𝑞_𝐼,Φ_𝐼) with its negative 𝑣∶=−𝑣.

(4)   Φ_𝐼-mesh quiver Γ(𝜕⁰_𝐼ℛ𝑞_𝐼,Φ_𝐼). By the equality Φ²_𝐼(𝑣)=𝑣−𝜕_𝐼(𝑣)⋅𝐡_𝐼, the Φ_𝐼-orbit of any 𝑣∈𝜕⁰_𝐼ℛ𝑞_𝐼 consists of two vectors 𝑣 and Φ_𝐼(𝑣). Now, we show that the Φ_𝐼-orbits in 𝜕⁰_𝐼ℛ𝑞_𝐼 form a Φ_𝐼-mesh translation quiver Γ(𝜕⁰_𝐼ℛ𝑞_𝐼,Φ_𝐼).

Note that Φ_𝐼(𝑒₅)=(−1,−1,1,1,−1), Φ²_𝐼(𝑒₅)=𝑒₅, 𝜕_𝐼(𝑒₅)=0, and 𝜕_𝐼(Φ_𝐼(𝑒₅))=0. It follows that the two-element Φ_𝐼-orbits of 𝑒₅ and −𝑒₅ lie in 𝜕⁰_𝐼ℛ𝑞_𝐼. Moreover, the vectors 𝑢₊⁽¹⁾=(1,1,−1,−1,−1),−𝑢₊⁽¹⁾,𝑤⁽⁰⁾=(),𝑤1,0,0,−1,0₊₍₀₎=Φ_𝐼𝑤⁽⁰⁾=(),𝑣0,1,−1,0,0₍₀₎=(),𝑣0,1,0,−1,0₊₍₀₎=Φ_𝐼𝑣⁽⁰⁾=()𝑤1,0,−1,0,0₊₍₁₎=(),𝑣2,1,−1,−2,0₊₍₁₎=(1,2,−1,−2,0) (96) belong to 𝜕⁰_𝐼ℛ𝑞_𝐼. It is easy to see that we have the following Φ²_𝐼-mesh quivers of vectors in 𝜕⁰_𝐼ℛ𝑞_𝐼: (97) Note that the Φ_𝐼-orbit of 𝑢₊⁽¹⁾ consists of the following two vectors: 𝑢⁽¹⁾=(2,−2,−2,−2,1),𝑢₊⁽¹⁾=Φ_𝐼𝑢⁽¹⁾. (98) By (92), a vector 𝑣=(𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝑣₅)∈ℤ⁵ is a root of 𝑞_𝐼∶ℤ⁵→ℤ of defect zero if and only if 2𝑣₁+𝑣₃+𝑣₄−𝑣₅²+2𝑣₂+𝑣₃+𝑣₄−𝑣₅²+2𝑣₃−𝑣₄²+2𝑣²₅𝑣=4,₁+𝑣₂+𝑣₃+𝑣₄=0. (99) It follows that 𝑣 or −𝑣 belongs to any of the six series of roots presented in Table 2.

Hence, we conclude that the Φ_𝐼-orbits in the set 𝜕⁰_𝐼ℛ𝑞_𝐼 form three Φ_𝐼-mesh quivers 𝒯_𝑢, 𝒯_𝑤, 𝒯_𝑣, and each of them has the form of infinite two-surface tube of rank 2: (100) where 𝑎 is one of the vectors 𝑢=𝑢⁽⁰⁾=(0,0,0,0,−1),𝑤=𝑤⁽⁰⁾=(1,0,0,−1,0),𝑣=𝑣⁽⁰⁾=(0,1,0,−1,0). (101)

(5)   Φ_𝐼-mesh quiver  Γ(𝜕⁰_𝐼ℛ𝑞_𝐼∪Ker𝑞_𝐼,Φ_𝐼). We recall that Ker𝑞_𝐼=ℤ⋅𝐡_𝐼, where 𝐡_𝐼=(1,1,−1,−1,0). Note that Φ_𝐼𝐡_𝐼=𝐡_𝐼,Φ_𝐼𝑚⋅𝐡_𝐼=𝑚⋅𝐡_𝐼,forany𝑚∈ℤ. (102) Obviously, the vectors lying in Ker𝑞_𝐼 form the Φ-mesh translation quiver 𝒯𝐡_𝐼 presented in (108).

Now, we construct from the Φ_𝐼-orbits in the set 𝜕⁰_𝐼ℛ𝑞_𝐼∪Ker𝑞_𝐼 an infinite Φ_𝐼-mesh translation quiver. For this purpose, we note that the following six vectors −𝑒₅,𝑒₅,Φ_𝐼−𝑒₅,Φ_𝐼𝑒₅,−𝐡_𝐼,𝐡_𝐼 (103) form two Φ_𝐼-meshes of width 1. If we complete them by the three vectors 0,𝑢₊⁽¹⁾∶=(1,1,−1,−1,−1),−𝑢₊⁽¹⁾, (104) we get the Φ_𝐼-mesh quiver (105) Analogously, we construct the following two Φ_𝐼-mesh quivers (106) where 𝑢⁽¹⁾=(),𝑢2,−2,−2,−2,1₊₍₁₎=Φ_𝐼𝑢⁽¹⁾,𝑤⁽⁰⁾=(),𝑤1,0,0,−1,0₊₍₀₎=Φ_𝐼𝑤⁽⁰⁾=(),𝑣0,1,−1,0,0₍₀₎=(),𝑣0,1,0,−1,0₊₍₀₎=Φ_𝐼𝑣⁽⁰⁾=(),𝑤1,0,−1,0,0₊₍₁₎=(),𝑣2,1,−1,−2,0₊₍₁₎=(1,2,−1,−2,0). (107) We recall that if 𝑣∈𝜕⁰_𝐼ℛ𝑞_𝐼, then 𝑣 or −𝑣 is one of the vectors presented in Table 2. It follows that the Φ_𝐼-orbits in 𝜕⁰_𝐼ℛ𝑞_𝐼∪ℤ⋅𝐡_𝐼 form three infinite Φ_𝐼-mesh sand-glass tubes 𝒯_𝑢, 𝒯_𝑤, 𝒯_𝑣 of rank (2,1), and each of them has the shape presented in (109) (108)(109) where 𝑎 is one of the vectors 𝑢=𝑢⁽⁰⁾=(0,0,0,0,−1),𝑤=𝑤⁽⁰⁾=(1,0,0,−1,0),𝑣=𝑣⁽⁰⁾=(0,1,0,−1,0). (110) Construct the disjoint union 𝒯_𝑢∪𝒯_𝑤∪𝒯_𝑣 of the tubes 𝒯_𝑢, 𝒯_𝑤, 𝒯_𝑣, and note that each of them contains the tube 𝒯𝐡_𝐼. By making the identification of the vectors 𝑚⋅𝐡_𝐼, with 𝑚∈ℤ, lying in the corresponding Φ_𝐼-orbits, we get the quotient Φ_𝐼-mesh translation quiver Γ𝜕⁰_𝐼ℛ𝑞_𝐼∪Ker𝑞_𝐼,Φ_𝐼=𝒯_𝑢∪𝒯_𝑤∪𝒯_𝑣≃ (111) that has a shape of a threefold sand-glass tube of rank (2,2,2,1) in the sense of [40]. It is obtained from the disjoint union of three copies of the onefold sand-glass tube of rank (2,1) presented in Figure 3 (see also [34, Figure 5.8]) by making an obvious identification of their waist vectors.

(6)   A  ℤ-congruence of the bigraph  Δ_𝐼  with the Euclidean diagram  𝔻^∼₄. Since we have cox_𝐼(𝑡)=𝐹^∼_𝔻4(𝑡)=𝑡⁵+𝑡⁴−2𝑡³−2𝑡²+𝑡+1 and 𝐬𝐩𝐞𝐜𝐜_𝐼=𝐬𝐩𝐞𝐜𝐜^∼_𝔻4, the Euclidean diagram 𝔻^∼₄ is the diagram 𝐷𝐼 associated to 𝐼. A technique developed in [2, 17, 18, 34, 40] allows us to construct a ℤ-invertible matrix 𝐵∈Gl(5,ℤ) such that the following diagrams are commutative: (112) where 𝑏^∼_𝔻4 and 𝑞^∼_𝔻4 are the forms of the Euclidean diagram (113) defined by the formulae 𝑏^∼_𝔻4(𝑥,𝑦)=𝑥₁𝑦₁+𝑥₂𝑦₂+𝑥₃𝑦₃+𝑥₄𝑦₄+𝑥₅𝑦₅−(𝑥₁+𝑥₂+𝑥₃+𝑥₄)𝑦₅=𝑥⋅𝐺^̌‌∼_𝔻4⋅𝑦^tr, 𝑞^∼_𝔻4(𝑥)=𝑏^∼_𝔻4(𝑥,𝑥), for 𝑥,𝑦∈ℤ⁵, ℎ_𝐵∶ℤ⁵→ℤ⁵ is the group automorphism defined by the formula ℎ_𝐵(𝑥)=𝑥⋅𝐵, and 𝐵=,10−1−1000−10−1000−1−101−1−1000111_̌‌𝐺^∼_𝔻4=.1000−10100−10010−10001−100001 (114) It is easy to check that the equality 𝐺^̌‌∼_𝔻4=𝐵⋅𝐶_𝐼⋅𝐵^tr holds, and therefore the diagrams (112) are commutative. Furthermore, by the same technique, we construct another matrix 𝐵₁=−1000−1000−1000−1000−100−111001 (115) such that the equality 𝐺^̌‌∼_𝔻4=𝐵₁⋅𝐶_𝐼⋅𝐵₁^tr holds.

6. Concluding Remarks

6.1 It follows from Lemma 3 and the results obtained recently in [3, 4] that for any connected positive (resp., principal) poset 𝐽, there exists a simply laced Dynkin diagram 𝐷𝐽∈{𝔸_𝑚,𝔻_𝑚,𝔼₆,𝔼₇,𝔼₈}  (resp., a simply laced Euclidean diagram 𝐷𝐽), uniquely determined by 𝐽, such that the symmetric Gram matrices 𝐺_𝐽, 𝐺_𝐷𝐽 are ℤ-congruent. Analogous Coxeter spectral classification of one-peak posets 𝐼, with almost 𝑃-critical Tits form 𝑞_𝐼∶ℤ^𝐼→ℤ, is obtained in [33] by a reduction to computer calculations.6.2. Although the Coxeter spectral classification problem for arbitrary finite posets remains unsolved, we have a solution for positive one-peak posets. Indeed, it follows from the results in [17] that for any one-peak positive poset 𝐽, there exists a simply laced Dynkin diagram 𝐷𝐽∈{𝔸_𝑠,𝔻_𝑛,𝔼₆,𝔼₇,𝔼₈}(uniquely determined by 𝐽) such that 𝐬𝐩𝐞𝐜𝐜_𝐽=𝐬𝐩𝐞𝐜𝐜_𝐷𝐽, the nonsymmetric Gram matrices 𝐺^̌‌_𝐽, 𝐺^̌‌_𝐷𝐽 are ℤ-congruent, and the symmetric Gram matrices 𝐺_𝐽, 𝐺_𝐷𝐽 are ℤ-congruent.6.3. We can determine the diagram 𝐷𝐽 as follows. Fix an upper-triangular numbering {𝑎₁,…,𝑎_𝑚} of elements of 𝐽. Then, the incidence matrix 𝐶_𝐽∈𝕄_𝑚(ℤ) is upper-triangular, and the Euler matrix 𝐶_𝐽∶=𝐶_𝐽⁻¹ is also upper triangular. Then, the Euler edge-bipartite graph Δ_𝐽 (33) is loop-free, and we have 𝐶_Δ𝐽=𝐶_𝐽. Hence, the symmetric Gram matrices 𝐺_Δ𝐽, 𝐺_𝐽 coincide, and, by Lemma 3, the poset 𝐽 is positive (resp., principal) if and only if the bigraph Δ_𝐽 is positive (resp., principal). By applying to Δ_𝐽 the inflation algorithm constructed in [4, 21] (see also [45]), we get (in a finite number of steps) an edge-bipartite graph 𝐷Δ_𝐽 such that the symmetric Gram matrix 𝐺_Δ𝐽=𝐺_𝐽 is ℤ-congruent with the symmetric Gram matrix 𝐺_𝐷Δ𝐽, and the edge-bipartite graph 𝐷Δ_𝐽 has no dotted edges; that is, 𝐷Δ_𝐽 is a (multi) graph. We set 𝐷𝐽∶=𝐷Δ_𝐽. It follows from the results in [3, 4] that 𝐷𝐽 is a simply laced Dynkin diagram, if 𝐽 is positive, and 𝐷𝐽 is a simply laced Euclidean diagram, if 𝐽 is principal. Moreover, the matrix 𝐺_𝐽 is ℤ-congruent with 𝐺_𝐷𝐽. Since the incidence Gram matrix 𝐺_𝐽 of 𝐽 is ℤ-congruent with the matrix 𝐺_𝐽 (by Proposition 5), then the matrices 𝐺_𝐽 and 𝐺_𝐷𝐽 are ℤ-congruent.6.4. Although we can apply in 6.3 the inflation algorithm to the incidence edge-bipartite graph Δ_𝐽, we use in the construction of 𝐷𝐽 the Euler edge-bipartite graph Δ_𝐽, because the number of nonzero entries in the Euler matrix 𝐶_𝐽∶=𝐶_𝐽⁻¹ does not increase the number for the matrix 𝐶_𝐽; see [28, Proposition 2.12]. It follows that the number of the dotted edges in Δ_𝐽 does not increase the number of the dotted edges in Δ_𝐽, and the use in 6.3 the bigraph Δ_𝐽 reduces the time of calculation in the procedure Δ_𝐽↦𝐷𝐽.

Acknowledgment

The research is supported by Polish Research Grant NCN 2011/03/B/ST1/00824.